Removability of the fundamental singularity for the heat equation and its consequences

Abdulla, Ugur G.

doi:10.1063/5.0233490

We prove the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the h-parabolic measure of the singularity point is null or positive. From the probabilistic point of view, the criterion presents an asymptotic law for conditional Brownian motion.

I. PRELUDE

Consider the fundamental solution of the heat equation:

F (x, t) = \{\begin{cases} {(4 π t)}^{- \frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}}, x \in R^{N}, t > 0, \\ 0, x \in R^{N}, x \neq 0, t = 0 . \end{cases}

(1)

It is a distributional solution of the Cauchty Problem

H F ≔ F_{t} - Δ F = 0 in R_{+}^{N + 1}, F = δ on R^{N} \times {t = 0}

where δ is a unit measure with support at x = 0. For any fixed

γ \in R^{N}

⁠, let

h (x, t) = F (x - γ, t)

be a fundamental solution with a pole at

O ≔ (γ, 0)

⁠. The singularity of h at

O

represents the natural phenomenon of the space-time distribution of the unit energy initially blown up at a single point. The fundamental singularity is non-removable for the heat equation in

R_{+}^{N + 1}

⁠. In particular, the Cauchy Problem for the heat equation in

R_{+}^{N + 1}

has infinitely many solutions in class O(h).

The goal of this paper is to reveal the criterion for the removability of the fundamental singularity for open subsets of

R_{+}^{N + 1}

⁠. Let

Ω \subset R_{+}^{N + 1}

be an arbitrary open set and

\partial Ω \cap {t = 0} = {O}

⁠. Let

g : \partial Ω \to R

be a boundary function such that g/h is a bounded Borel measurable. Consider a singular parabolic Dirichlet problem (PDP):

H u = 0 in Ω, u = g on \partial Ω \ {O}; u = O (h) at O .

(2)

Solution of the singular PDP is understood in the Perron’s sense [see Sec. II A, and formulae and ]. Furthemore, the expression “prescribing the behavior of u/h at

O

” is understood in the sense of requiring relations and at the boundary point

O

[similar convention is made for the singular PDP ]. Without prescribing the behavior of u/h at

O

⁠, either there exists one and only one or infinitely many solutions of PDP [see Sec. II B and formula ]. The main goal of this paper is to find a necessary and sufficient condition for open sets Ω for the uniqueness of the solution to the PDP without prescribing u/h at

O

⁠. The problem of removability vs non-removability of the fundamental singularity is equivalent to the question of the uniqueness of the solution to PDP without prescribing the behavior of u/h at

O

⁠. Note that we are not requiring existence of the limit of u/h at

O

⁠. Assume that

\lim_{z \to O, z \in \partial Ω \ {O}} g / h

exists. We prove that in this case the removability of the fundamental singularity at

O

is equivalent to the existence of a unique solution of the singular PDP such that

\lim_{z \to O, z \in Ω} \frac{u}{h} = \lim_{z \to O, z \in \partial Ω \ {O}} \frac{g}{h} .

Otherwise speaking, for a unique solution u, u/h will pick up the limit value at the singularity point

O

without being required (see Sec. II C and Definition II.6).

The following procedure provides a key problem in testing the removability of fundamental singularity. Let

Ω_{n} ≔ Ω \cap {t > n^{- 1}}, n = 1, 2, \dots

and u_n be a unique solution of the parabolic Dirichlet problem

H u = 0, in Ω_{n}; u |_{\partial Ω_{n} \cap {t > n^{- 1}}} = 0; u |_{\partial Ω_{n} \cap {t = n^{- 1}}} = h (x, n^{- 1}) .

(3)

From the maximum principle it follows that

0 \leq u_{n + 1} (x, t) \leq u_{n} (x, t) \leq h (x, t), on Ω_{n} .

(4)

Therefore, there exists a limit function

u_{*} (x, t) = \lim_{n \to + \infty} u_{n} (x, t), (x, t) \in Ω,

(5)

which satisfies , and

0 \leq u_{*} (x, t) \leq h (x, t), (x, t) \in Ω .

The following is the key problem.

Problem $A_{γ}$ ⁠: Is u_∗ ≡ 0 in Ω or $u_{*} (x, t) ≢ 0$ in Ω? Equivalently, is the fundamental singularity at $O$ removable or nonremovable for Ω?

Note that the Problems $A_{γ}$ are equivalent for various γ. Indeed, the heat equation is translation invariant so that solving the problem when the pole of the fundamental solutions is (0, 0) is equivalent to solve the problem when the pole is at (γ, 0) for any $γ \in R^{N}$ ⁠.

Next, we formulate the equivalent problem in

R_{-}^{N + 1}

⁠. In that context, we are going to consider one-point Alexandrov compactification:

R_{-}^{N + 1} \to R_{-}^{N + 1} \cup {\infty}

⁠. For any fixed finite

γ \in R^{N}

⁠, consider a function

\tilde{h} (x, t) = e^{⟨ x, γ ⟩ + | γ |^{2} t} .

(6)

It is a positive solution of the heat equation in

R_{-}^{N + 1}

⁠. If γ = 0, then

\tilde{h} \equiv 1

⁠, while in the case when γ ≠ 0, it is an unbounded solution with singularity at ∞. The singularity is not removable for the heat equation in

R_{-}^{N + 1}

⁠. We aim to reveal the criterion for the removability of the fundamental singularity for the open subsets of

R_{-}^{N + 1}

⁠.

Let

\tilde{Ω} \subset R_{-}^{N + 1}

be an arbitrary open set, and

g : \partial \tilde{Ω} \to R

be a boundary function, such that

g / \tilde{h}

is a bounded Borel measurable. Consider a singular parabolic Dirichlet problem (PDP):

H u = 0 in \tilde{Ω}, u = g on \partial \tilde{Ω}; u = O (\tilde{h}) at \infty,

(7)

Without prescribing the behavior of

u / \tilde{h}

at ∞, there exists one and only one or infinitely many solutions of [see Sec. II B, formula ]. The alternative statement is the question of removability vs non-removability of the fundamental singularity at ∞ for

\tilde{Ω}

⁠. In particular, in the case γ = 0

(\tilde{h} \equiv 1)

⁠, we are addressing the uniqueness of a bounded solution of , the problem solved in Ref. 1. Similar to its counterpart , the key problem to test the removability of the singularity at ∞ is formulated as follows:

Let

{\tilde{Ω}}_{n} ≔ \tilde{Ω} \cap {t > - n}, n = 1, 2, \dots

and

{\tilde{u}}_{n}

be a unique solution of the parabolic Dirichlet problem

H u = 0, in {\tilde{Ω}}_{n}; u |_{\partial {\tilde{Ω}}_{n} \cap {t > - n}} = 0; u |_{\partial {\tilde{Ω}}_{n} \cap {t = - n}} = \tilde{h} (x, - n) .

(8)

From the maximum principle it follows that

0 \leq {\tilde{u}}_{n + 1} (x, t) \leq {\tilde{u}}_{n} (x, t) \leq \tilde{h} (x, t), on {\tilde{Ω}}_{n} .

(9)

Therefore, there exists a limit function

{\tilde{u}}_{*} (x, t) = \lim_{n \to + \infty} {\tilde{u}}_{n} (x, t), (x, t) \in \tilde{Ω},

(10)

which satisfies , and

0 \leq {\tilde{u}}_{*} (x, t) \leq \tilde{h} (x, t), (x, t) \in \tilde{Ω} .

The following is the key problem.

Problem ${\tilde{A}}_{γ}$ ⁠: Is ${\tilde{u}}_{*} \equiv 0$ in $\tilde{Ω}$ or ${\tilde{u}}_{*} (x, t) ≢ 0$ in $\tilde{Ω}$ ? Equivalently, is a fundamental singularity at ∞ removable or nonremovable for $\tilde{Ω}$ ?

Remark I.1.

Problem ${\tilde{A}}_{γ}$ can be formulated in $R^{N + 1}$ without any change. Indeed, the parabolic Dirichlet problem for the heat equation is uniquely solvable in any open subset $R_{+}^{N + 1}$ in a class $O (\tilde{h})$ ⁠. Therefore, given arbitrary open set $Ω \subset R^{N + 1}$ , the solution of the parabolic Dirichlet problem in Ω can be constructed as a unique continuation of the solution to the parabolic Dirichlet problem in $Ω_{-} = Ω \cap R_{-}^{N + 1}$ . Moreover, the latter is independent of the boundary values assigned on ∂Ω₋∩{t = 0}, since ∂Ω₋∩{t = 0} is a parabolic measure null subset of ∂Ω₋. This implies that the Problem ${\tilde{A}}_{γ}$ is equivalent for Ω and Ω₋. Otherwise speaking, the fundamental singularity at ∞ is removable for $Ω \subset R^{N + 1}$ if and only if it is removable for Ω₋.

The only problem in the family of formulated problems that is solved is the Problem

{\tilde{A}}_{γ}

when γ = 0 (or Problem

{\tilde{A}}_{0}

⁠). The Problem

{\tilde{A}}_{0}

was formulated by Kolmogorov in 1928 in the seminar on the probability theory at Moscow State University in the particular case with

Ω = {| x | < f (t), - \infty < t < 0} \subset R^{2}

⁠, with f ∈ C(−∞, 0] such that

f (- \infty) = + \infty, f ↑ + \infty, {(- t)}^{- \frac{1}{2}} f ↑ + \infty

as t↓ − ∞. Kolmogorov’s motivation for posing this problem was a connection to the probabilistic problem of finding asymptotic behavior at infinity of the standard Brownian path. Let {ξ(t): t ≥ 0, P_●} be a standard one-dimensional Brownian motion and P_●(B) is the probability of the event B as a function of the starting point ξ(0). Consider the event

B = [the set {t \geq 0 : ξ (t) > f (- t)} clusters to + \infty]

Blumenthal’s 01 law implies that P₀(B) = 0 or 1; f(−t) is said to belong to the lower class if this probability is 1 and to the upper class otherwise. Remarkably, Kolmogorov Problem’s solution u_∗ is = 0 or

> 0

according to as f(−t) is in lower or upper class accordingly. Kolmogorov Problem in a one-dimensional setting was solved by Petrovsky in 1935, and the celebrated result is called the Kolmogorov–Petrovski test in the probabilistic literature² (see also Ref. 3).

The full solution of the Kolmogorov Problem for arbitrary open sets Ω (or Problem ${\tilde{A}}_{0}$ ⁠) is presented in Ref. 1. A new concept of regularity or irregularity of ∞ is introduced according to whether the parabolic measure of ∞ is null or positive, and the necessary and sufficient condition for the Problem ${\tilde{A}}_{0}$ is proved in terms of the Wiener-type criterion for the regularity of ∞.

In the probabilistic context, the formulated problems $A_{γ}$ and ${\tilde{A}}_{γ}$ are generalizations of the Kolmogorov problem to establish asymptotic laws for the h-Brownian processes.⁴

A. Overture: Kolmogorov–Petrovsky-type test for the removability of the fundamental singularity

For the special case of domains

Ω = {(x, t) : | x - γ | < l (t), t > 0} \subset R_{+}^{N + 1},

(11)

where

l \in C ({\bar{R}}_{+}; {\bar{R}}_{+}), l (0) = 0, l (t) > 0

for t > 0, the solution of the Problem

A_{γ}

reads:

Theorem I.2.

u_∗ ≡ 0 or u_∗ > 0, that is to say, the fundamental singularity is removable or non-removable according to whether the following integral diverges or converges

\int_{0^{+}} t^{- \frac{N}{2} - 1} l^{\frac{N}{2}} (t) e^{- \frac{l}{4 t}} d t

(12)

The result is local. The removability of the fundamental singularity is dictated by the boundary of the domain near the singularity point. Precisely, it is defined by the thinness of the exterior set $R_{+}^{N + 1} - Ω$ near the singularity point $O$ ⁠.

The removability of the singularity is locally order-preserving. Precisely, if for some δ > 0 we have Ω₁ ∩ {0 < t < δ} ⊂ Ω₂ ∩ {0 < t < δ}, then removability of the fundamental singularity for Ω₂ (or non-removability for Ω₁) implies the same for Ω₁ (or Ω₂) (Lemma IV.2, Sec. II C).

An equivalent form of the criterion can be written if we choose

l (t) = {(4 t \log ρ (t))}^{\frac{1}{2}}

⁠, and consider the domain Ω such that

Ω \cap {0 < t < δ} = {(x, t) : | x - γ |^{2} < 4 t \log ρ (t), 0 < t < δ}

(13)

such that

ρ \in C (0, δ), ρ > 0; ρ (t) ↑ + \infty, t \log ρ (t) \to 0, as t ↓ 0 .

(14)

Then the claim of the Theorem I.2 remains valid if the integral is replaced with

\int_{0^{+}} \frac{| \log ρ (t) |^{\frac{N}{2}}}{t ρ (t)} d t

(15)

Some examples of functions ρ with divergent integral are as follows:

| \log t |, | \log t | \cdot \log_{2}^{\frac{N}{2} + 1} t, | \log t | \cdot \log_{2}^{\frac{N}{2} + 1} t \cdot \log_{3} t \dots \log_{k} t, k = 3, 4, \dots

(16)

On the other side, for ∀ϵ > 0, the integral converges for the corresponding functions

| \log t |^{1 + ϵ}, | \log t | \cdot \log_{2}^{\frac{N}{2} + 1 + ϵ} t, | \log t | \cdot \log_{2}^{\frac{N}{2} + 1} t \cdot \log_{3} t \dots \log_{k}^{1 + ϵ} t, k = 3, 4, \dots

(17)

We adopt the notation

\log_{2} t = \log | \log t |, \log_{k} t = \log \log_{k - 1} t, k = 3, 4, \dots

Hence, we have the following law for the removability of the fundamental singularity. For arbitrary integer k ≥ 4, consider a domain

| x - γ |^{2} \leq 4 t [\log_{2} t + (\frac{N}{2} + 1) \log_{3} t + \dots + \log_{k}^{1 + ϵ} t]

(18)

Then u_∗ ≡ 0 or u_∗ > 0, that is to say, the fundamental singularity is removable or non-removable according to ϵ = 0 or ϵ > 0.

Probabilistic counterpart: Let {x(t) = (x₁(t), …, x_N(t)): t ≥ 0, P_●} be an N-dimensional h-Brownian process, and P_●(B) is a probability of the event B as a function of the starting point x(τ) with τ > 0.⁴ The process x(t) is an almost surely continuous process whose sample functions never leave

R_{+}^{N + 1}

and proceed downward, that is, in the direction of decreasing t. In fact, almost every path starting at x(τ) has a finite lifetime τ and tends to the boundary point

O

as t↓ 0.⁴ Let r(t) = |x(t) − γ|: 0 < t ≤ τ be a radial part of the h-Brownian path starting at x(τ). Consider the event

B = [the set {0 < t \leq τ : r (t) > l (t)} clusters to 0]

Kolmogorov’s 01 law implies that P_x(τ)(B) = 0 or 1; l is said to belong to the lower class if the probability is 1 and to the upper class otherwise. The probabilistic analog of Theorem I.2 states that if l ∈ ↑ and if t^−1/2l ∈ ↓ for small t > 0, then l belongs to the upper class or to the lower class according as the integral converges or diverges. In particular, for arbitrary integer k ≥ 4

l (t) = 2 t^{\frac{1}{2}} {[\log_{2} t + (\frac{N}{2} + 1) \log_{3} t + \dots + \log_{k}^{1 + ϵ} t]}^{\frac{1}{2}}

belongs to the upper or lower class according as ϵ > 0 or ϵ ≤ 0.

Next, we describe the solution of the Problem

{\tilde{A}}_{γ}

for a special class of domains

\tilde{Ω} = {(x, t) : | x + 2 t γ |^{2} < - 4 t \log ρ (t), - \infty < t < δ} \subset R_{-}^{N + 1},

(19)

where δ ≪ − 1 and

ρ \in C (- \infty, δ), ρ > 0, ρ (t) \to + \infty, t^{- 1} \log ρ (t) \to 0, as t ↓ - \infty .

(20)

Theorem I.3.

{\tilde{u}}_{*} \equiv 0

or

{\tilde{u}}_{*} > 0

, that is to say the singularity at ∞ is removable or non-removable according to the following integral diverges or converges

\int_{- \infty} \frac{| \log ρ (t) |^{\frac{N}{2}}}{t ρ (t)} d t

(21)

Typical examples for the divergence or the convergence of the integral are given by and just by replacing t with |t|. Hence, we have the following law for the removability of singularity at ∞. For arbitrary integer k ≥ 4, consider a domain with

| x + 2 t γ |^{2} < - 4 t [\log_{2} | t | + (\frac{N}{2} + 1) \log_{3} | t | + \dots + \log_{k}^{1 + ϵ} | t |]

Then u_∗ ≡ 0 or u_∗ > 0, that is to say the singularity at ∞ is removable or non-removable according to ϵ ≤ 0 or ϵ > 0.

Remarkably, in the particular case γ = 0, Theorem I.3 coincides with the celebrated Kolmogorov–Petrovski test.^2,3

Probabilistic counterpart: Let {x(t) = (x₁(t), …, x_N(t)): t < 0, P_●} be an N-dimensional

\tilde{h}

-Brownian process, and P_●(B) is a probability of the event B as a function of the starting point x(τ) with τ < 0.⁴ The process x(t) is an almost surely continuous process whose sample functions never leave

R_{-}^{N + 1}

and proceed downward, that is, in the direction of decreasing t. Almost every path starting at x(τ) tends to the boundary point ∞ as t↓ − ∞.⁴ Let r(t) = |x(t) + 2tγ|: t < 0 be the radial part of the

\tilde{h}

-Brownian path starting at x(τ). Consider the event

B = [the set {- \infty < t \leq τ : r (t) > l (t)} clusters to - \infty]

Kolmogorov’s 01 law implies that P_x(τ)(B) = 0 or 1; l is said to belong to the lower class if the probability is 1 and to the upper class otherwise. The probabilistic analog of Theorem I.3 states that if ρ satisfies , then

l (t) = 2 {(- t \log ρ (t))}^{\frac{1}{2}}

belongs to the upper class or the lower class according as the integral converges or diverges. In particular, for arbitrary integer k ≥ 4

l (t) = 2 | t |^{\frac{1}{2}} {[\log_{2} | t | + (\frac{N}{2} + 1) \log_{3} | t | + \dots + \log_{k}^{1 + ϵ} | t |]}^{\frac{1}{2}}

belongs to the upper or lower class according as ϵ > 0 or ϵ ≤ 0.

II. FORMULATION OF PROBLEMS

Being a generalization of the Kolmogorov problem, the Problems $A_{γ}$ ⁠, ${\tilde{A}}_{γ}$ ⁠, and their solution expressed in Theorems I.2 and I.3 has far-reaching measure-theoretical, topological and probabilistic implications in Analysis, PDEs and Potential theory. The goal of this section is to formulate three outstanding problems equivalent to the Problems $A_{γ}$ and ${\tilde{A}}_{γ}$ ⁠. Since the problems $A_{γ}$ and ${\tilde{A}}_{γ}$ are equivalent via the Appell transformation, without loss of generality we are going to formulate the problems in the framework of the Problem $A_{γ}$ ⁠. The equivalent formulation can be pursued in the framework of the Problem ${\tilde{A}}_{γ}$ by replacing the triple $(R_{+}^{N + 1}, Ω, h)$ with singularity point at $O$ through the triple $(R_{-}^{N + 1}, \tilde{Ω}, \tilde{h})$ with the singularity point at ∞ respectively.

A. Unique solvability of the singular parabolic Dirichlet problem

Consider a singular parabolic Dirichlet problem(PDP) (2). The solution of the PDP can be constructed by Perron’s method (or the method by Perron, Wiener, Brelot, and Bauer).^4,5 Let us introduce some necessary terminology.

We will often write a typical point

z \in R^{N + 1}

as

z = (x, t), x \in R^{N}, t \in R

⁠. A smooth solution of the heat equation is called a parabolic function. A bounded open set

U \subset R^{N + 1}

is regular if for each continuous

f : \partial U \to R

there exists one (and only one) parabolic function

H_{f}^{U} : U \to R

⁠, such that

\lim_{\begin{array}{c} z \to w \\ w \in \partial U \end{array}} H_{f}^{U} (z) = f (w), w \in \partial U .

A function u is called a superparabolic in Ω if it satisfies the following conditions:

−∞ < u ≤ +∞, u < +∞ on a dense subset of Ω;
u is lower semicontinuous (l.s.c.);
for each regular open set U ⊂ Ω and each parabolic function $v \in C (\bar{U})$ ⁠, the inequality u ≥ v on ∂U implies u ≥ v in U.

A function u is called a subparabolic if −u is a superparabolic. We use the notation $S (Ω)$ for a class of superparabolic functions. Similarly, the class of subparabolic functions is $- S (Ω)$ ⁠.

A function u = v/h is called a h-parabolic, h-superparabolic, or h-subparabolic in Ω if v is parabolic, superparabolic, or subparabolic.⁴

We use the notation $S_{h} (Ω)$ [respectively $- S_{h} (Ω)$ ] for a class of all h-superparabolic (respectively h-subparabolic) functions in Ω.

Given boundary function f on ∂Ω, consider a h-parabolic Dirichlet problem (h-PDP): find h-parabolic function u in Ω such that

u = f on \partial Ω

(22)

It is easy to see that h-parabolic function

u = \frac{v}{h}

is a bounded solution of the h-PDP if and only if v is a solution of the PDP .

Assuming for a moment that f ∈ C(∂Ω), the generalized upper (or lower) solution of the h-PDP is defined as

{}^{h}{\bar{H}}_{f}^{Ω} \equiv \inf {u \in S_{h} (Ω) : \underset{z \to w, z \in Ω}{lim inf} u \geq f (w) for all w \in \partial Ω}

(23)

{}^{h}{\underset{̲}{H}}_{f}^{Ω} \equiv \sup {u \in - S_{h} (Ω) : \underset{z \to w, z \in Ω}{lim sup} u \leq f (w) for all w \in \partial Ω}

(24)

The class of functions defined in [or in ] is called upper class (or lower class) of the h-PDP. According to classical potential theory,⁴ f is a h-resolutive boundary function in the sense that

{}^{h}{\bar{H}}_{f}^{Ω} \equiv {}^{h}{\underset{̲}{H}}_{f}^{Ω} \equiv {}^{h}H_{f}^{Ω} .

The indicator function of any Borel measurable boundary subset, and equivalently any bounded Borel measurable boundary function is resolutive. Being h-parabolic in Ω,

{}^{h}H_{f}^{Ω}

is called a generalized solution of the h-PDP for f. The generalized solution is unique by construction. It is essential to note that the construction of the generalized solution is accomplished by prescribing the behavior of the solution at

O

⁠.

Equivalently, we can define a generalized solution of the PDP (2):

Definition II.1.

Let

g : \partial Ω \to R

be a boundary function, such that g/h is a bounded Borel measurable. Then g is called a resolutive boundary function for the PDP (2), if f = g/h (extended to

O

) is h-resolutive for the h-PDP. The function

H_{g}^{Ω} ≔ h {}^{h}H_{f}^{Ω}

(25)

is called a generalized solution of the PDP (2).

In particular, for the resolutive boundary function g we have

{\bar{H}}_{g}^{Ω} \equiv {\underset{̲}{H}}_{g}^{Ω} \equiv H_{g}^{Ω},

where

{\bar{H}}_{g}^{Ω} \equiv \inf \{u \in S (Ω) : \underset{z \to w, z \in Ω}{lim inf} \frac{u}{h} \geq \underset{z \to w, z \in \partial Ω}{lim inf} \frac{g}{h} for all w \in \partial Ω\}

(26)

{\underset{̲}{H}}_{g}^{Ω} \equiv \sup \{u \in - S (Ω) : \underset{z \to w, z \in Ω}{lim sup} \frac{u}{h} \leq \underset{z \to w, z \in \partial Ω}{lim sup} \frac{g}{h} for all w \in \partial Ω\}

(27)

Again, note that the unique solution $H_{g}^{Ω}$ of the PDP (2) is constructed by prescribing the behavior of the ratio $H_{g}^{Ω} / h$ at $O$ ⁠.

The elegant theory, while identifying a class of unique solvability, leaves the following questions open:

Would a unique solution of the h-PDP still exist if its limit at $O$ were not specified? That is, could it be that the solutions would pick up the “boundary value” $f (O)$ without being required? Equivalently, would a unique solution of the PDP (2) still exist if the limit of the ratio of solution to h at $O$ is not prescribed? In particular, is the fundamental singularity at $O$ removable?
What if the boundary datum f (or g/h) on ∂Ω, while being continuous at $\partial Ω \ {O}$ ⁠, does not have a limit at $O$ ⁠, for example, it exhibits bounded oscillations. Is the h-PDP [or PDP (2)] uniquely solvable?

Example II.2.

Let

Ω = R_{+}^{N + 1}

. It is easy to see that the boundary of

R_{+}^{N + 1}

is h-resolutive and the only possible solutions of the h-PDP in

R_{+}^{N + 1}

are constants. Precisely, the unique solution of the h-PDP is identical with the constant

f (O)

. Indeed, for arbitrary ϵ > 0, the function

u (\cdot) = f (O) + \frac{ϵ}{h (\cdot)} (or v (\cdot) = f (O) - \frac{ϵ}{h (\cdot)})

is in the upper class (or lower class) for h-PDP in

R_{+}^{N + 1}

for f. Hence,

f (O) - \frac{ϵ}{h (\cdot)} \leq {}^{h}{\underset{̲}{H}}_{f}^{R_{+}^{N + 1}} (\cdot) \leq {}^{h}{\bar{H}}_{f}^{R_{+}^{N + 1}} (\cdot) \leq f (O) + \frac{ϵ}{h (\cdot)}

Since ϵ > 0 is arbitrary, the assertion follows. Equivalently, all possible solutions of the PDP (2) in

R_{+}^{N + 1}

are constant multiples of h, and the unique solution is identified by prescribing the ratio u/h at

O

⁠.

Example II.2 demonstrates that if $Ω = R_{+}^{N + 1}$ ⁠, the answer is negative and arbitrary constant C is a solution of the h-PDP, Ch is a solution of the PDP (2), and the fundamental singularity at $O$ is not removable.

The positive answer to these fundamental questions is possible if Ω is not too sparse, or equivalently $Ω^{c} \cap R_{+}^{N + 1}$ is not too thin near $O$ ⁠. The principal purpose of this paper is to prove the necessary and sufficient condition for the non-thinness of $Ω^{c} \cap R_{+}^{N + 1}$ near $O$ which is equivalent to the uniqueness of the solution of the h-PDP [or PDP (2)] without specification of the boundary function (or ratio of the boundary function to h) at $O$ ⁠.

Furthermore, given bounded Borel measurable function $f = g / h : \partial Ω \ {O} \to R$ ⁠, we fix an arbitrary finite real number $\bar{f}$ ⁠, and extend a function f as $f (O) = \bar{f}$ ⁠. The extended function is a bounded Borel measurable on ∂Ω and there exists a unique solution ${}^{h}H_{f}^{Ω}$ of the h-PDP, and the unique solution of the PDP (2) is given by (25). The major question now becomes:

Problem 1.

How many bounded solutions do we have, or does the constructed solution depend on $\bar{f}$ ?

B. Characterization of the h-parabolic measure of singularity point

For a given boundary Borel subset A ⊂ ∂Ω, denote the indicator function of A as 1_A. Indicator functions of the Borel measurable subsets of ∂Ω are resolutive.⁴^, h-Parabolic measure of the boundary Borel subset A is defined as⁴:

μ_{Ω}^{h} (z, A) = {}^{h}H_{1_{A}}^{Ω} (z),

where z ∈ Ω is a reference point. It is said that A is an h-parabolic measure null set if μ_Ω(·, A) vanishes identically in Ω. If this is not the case, A is a set of positive h-parabolic measure. In particular, the h-parabolic measure of

{O}

is well defined:

μ_{Ω}^{h} (\cdot, {O}) = {}^{h}H_{1_{{O}}}^{Ω} (\cdot) .

The following formula is true for the solution

{}^{h}H_{f}^{Ω}

of the h-PDP:⁴

{}^{h}H_{f}^{Ω} (z) = \int_{\partial Ω} f (w) μ_{Ω}^{h} (z, d w), z \in Ω

(28)

Since f is extended to

{O}

as

f (O) = \bar{f}

⁠, we have

{}^{h}H_{f}^{Ω} (z) = \int_{\partial Ω \ {O}} f (w) μ_{Ω}^{h} (z, d w) + \bar{f} \cdot {}^{h}H_{1_{{O}}}^{Ω} (z), z \in Ω

(29)

This formula implies that the uniqueness of the solution to the h-PDP without prescribing the behavior of the solution at the singularity point

O

⁠, that is to say, the independence of

{}^{h}H_{f}^{Ω}

on

\bar{f}

is equivalent to whether or not

O

is an h-parabolic measure null set. Equivalently, according to the formula the following formula is true for the unique solution of the PDP :

H_{g}^{Ω} (z) = h (z) \int_{\partial Ω} \frac{g (w)}{h (w)} μ_{Ω}^{h} (z, d w), z \in Ω

(30)

Splitting the integral as in we have

H_{g}^{Ω} (z) = h (z) \int_{\partial Ω \ {O}} \frac{g (w)}{h (w)} μ_{Ω}^{h} (z, d w) + \bar{f} h (z) {}^{h}H_{1_{{O}}}^{Ω} (z), z \in Ω .

(31)

where

\bar{f}

is a prescribed limit value of

H_{g}^{Ω} / h

at

O

⁠. Similar to its counterpart , the formula demonstrates that the uniqueness of the solution u of the PDP without prescribing u/h at

O

is equivalent to whether or not

O

is an h-parabolic measure null set.

Hence, the following problem is the measure-theoretical counterpart of the Problem 1:

Problem 2.

Given Ω, is the h-parabolic measure of ${O}$ null or positive ?

From the Example II.2 demonstrated above it follows that in the particular case with

Ω = R_{+}^{N + 1}

⁠, we have

μ_{R_{+}^{N + 1}}^{h} (\cdot, {O}) \equiv 1, μ_{R_{+}^{N + 1}}^{h} (\cdot, \partial R_{+}^{N + 1} - {O}) \equiv 0 .

(32)

Example II.3.

For arbitrary c > 0 consider a domain bounded by the level set of h:

Ω = {h > c^{'}} = \{z : | x - γ |^{2} < - 2 N t \log \frac{t}{c}, 0 < t < c\},

where

c^{'} = {(4 π c)}^{- \frac{N}{2}}

. It is easy to see that the h-parabolic measure of

{O}

is positive, and we have

{}^{h}H_{1_{{O}}}^{Ω} (z) = \frac{h (z) - c^{'}}{h (z)}, z \in Ω .

(33)

Both Problem 1 and 2 are equivalent to the Problem

A_{γ}

formulated in Sec. I. The connection follows from the following formula:

u_{*} (z) = h (z) {}^{h}H_{1_{{O}}}^{Ω} (z), z \in Ω

(34)

To establish , first note that the h-parabolic function u_∗/h is in the lower class of the Perron’s solution

{}^{h}H_{1_{{O}}}^{Ω}

⁠, which imply that

\frac{u_{*} (z)}{h (z)} \leq {}^{h}H_{1_{{O}}}^{Ω} (z), z \in Ω .

(35)

Moreover,

{}^{h}H_{1_{{O}}}^{Ω}

itself is in the lower class of the Perron’s solution u_n/h of the h-PDP in Ω_n with boundary function

1_{\partial Ω_{n} \cap {t = n^{- 1}}}

⁠, where u_n is a solution of PDP . Therefore, we have

{}^{h}H_{1_{{O}}}^{Ω} (z) \leq \frac{u_{n} (z)}{h (z)}, z \in Ω_{n} .

(36)

passing to the limit as n → ∞, from , , , and follows.

In light of the measure-theoretical counterpart of the removability of the fundamental singularity, we introduce a concept of h-regularity of the boundary point $O$ ⁠.

Definition II.4.

$O$ is said to be h-regular for Ω if it is an h-parabolic measure null set. Conversely, $O$ is h-irregular if it has a positive h-parabolic measure.

Hence, Theorem I.2 establishes a criterion for the removability of the fundamental singularity in terms of the necessary and sufficient condition for the h-regularity of $O$ ⁠.

Similarly, in the context of the singular PDP , and corresponding

\tilde{h}

-PDP, we have the formulae analogous to , , and :

{}^{\tilde{h}}H_{f}^{\tilde{Ω}} (z) = \int_{\partial \tilde{Ω} \ {\infty}} f (w) μ_{\tilde{Ω}}^{\tilde{h}} (z, d w) + \bar{f} \cdot {}^{\tilde{h}}H_{1_{{\infty}}}^{\tilde{Ω}} (z), z \in \tilde{Ω}

(37)

H_{g}^{\tilde{Ω}} (z) = \tilde{h} (z) \int_{\partial \tilde{Ω} \ {\infty}} \frac{g (w)}{h (w)} μ_{\tilde{Ω}}^{\tilde{h}} (z, d w) + \bar{f} \tilde{h} (z) {}^{\tilde{h}}H_{1_{{\infty}}}^{\tilde{Ω}} (z), z \in \tilde{Ω}

(38)

{\tilde{u}}_{*} (z) = \tilde{h} (z) {}^{\tilde{h}}H_{1_{{\infty}}}^{\tilde{Ω}} (z), z \in \tilde{Ω} .

(39)

We introduce a concept of

\tilde{h}

-regularity of the boundary point ∞ for

\tilde{Ω} \subset R_{-}^{N + 1}

⁠.

Definition II.5.

∞ is said to be $\tilde{h}$ -regular for $\tilde{Ω}$ if it is an $\tilde{h}$ -parabolic measure null set. Conversely, ∞ is $\tilde{h}$ -irregular if it has a positive $\tilde{h}$ -parabolic measure.

In fact, in the particular case with $γ = 0, \tilde{h} \equiv 1$ ⁠, it coincides with the concept of regularity of ∞ introduced in Ref. 1. Theorem I.3 presents a criterion for the removability of the fundamental singularity at ∞ in terms of the necessary and sufficient condition for the $\tilde{h}$ -regularity of ∞.

C. Boundary regularity in singular Dirichlet problem

The notion of the h-regularity of $O$ is, in particular, relates to the notion of continuity of the solution to the h-PDP at $O$ ⁠.

Problem 3.

Given Ω, determine whether or not

\underset{z \to O, z \in \partial Ω}{lim inf} f \leq \underset{z \to O, z \in Ω}{lim inf} {}^{h}H_{f}^{Ω} \leq \underset{z \to O, z \in Ω}{lim sup} {}^{h}H_{f}^{Ω} \leq \underset{z \to O, z \in \partial Ω}{lim sup} f,

(40)

for all bounded

f \in C (\partial Ω \ {O})

⁠.

Note that if f has a limit at $O$ ⁠, (40) simply means that the solution ${}^{h}H_{f}^{Ω}$ is continuous at $O$ ⁠.

The equivalent problem in the context of the PDP (2) is the following:

Problem 3′.

Given Ω, whether or not

\underset{z \to O, z \in \partial Ω}{lim inf} \frac{g}{h} \leq \underset{z \to O, z \in Ω}{lim inf} \frac{H_{g}^{Ω}}{h} \leq \underset{z \to O, z \in Ω}{lim sup} \frac{H_{g}^{Ω}}{h} \leq \underset{z \to O, z \in \partial Ω}{lim sup} \frac{g}{h},

(41)

for all g such that

\frac{g}{h} \in C (\partial Ω \ {O})

and bounded.

In particular, if g/h has a limit at $O$ ⁠, (41) means that the limit of the ratio $H_{g}^{Ω} / h$ at $O$ exists and equal to the limit of g/h.

Definition II.6.

Singular PDP (2) (and corresponding h-PDP) is said to be regular at $O$ if (40) and (41) are satisfied. It is said to be irregular at $O$ otherwise.

Similarly, we introduce the concept of regularity at ∞ of the singular PDP , and corresponding

\tilde{h}

-PDP if the following conditions are satisfied:

\begin{align} \underset{z \to \infty, z \in \partial \tilde{Ω}}{lim inf} f & \leq \underset{z \to \infty, z \in Ω}{lim inf} {}^{\tilde{h}}H_{f}^{\tilde{Ω}} \leq \underset{z \to \infty, z \in \tilde{Ω}}{lim sup} {}^{\tilde{h}}H_{f}^{\tilde{Ω}} \leq \underset{z \to \infty, z \in \partial \tilde{Ω}}{lim sup} f, \\ \forall bounded f \in C (\partial \tilde{Ω}) \end{align}

(42)

\begin{align} \underset{z \to \infty, z \in \partial \tilde{Ω}}{lim inf} \frac{g}{\tilde{h}} & \leq \underset{z \to \infty, z \in \tilde{Ω}}{lim inf} \frac{H_{g}^{\tilde{Ω}}}{\tilde{h}} \leq \underset{z \to \infty, z \in \tilde{Ω}}{lim sup} \frac{H_{g}^{\tilde{Ω}}}{\tilde{h}} \leq \underset{z \to \infty, z \in \partial \tilde{Ω}}{lim sup} \frac{g}{\tilde{h}}, \\ \forall bounded \frac{g}{\tilde{h}} \in C (\partial \tilde{Ω}) \end{align}

(43)

Theorem I.2 (or I.3) express the solutions to equivalent Problems 1–3 in terms of the Kolmogorov–Petrovsky-type criterion for the h-regularity of

O

(or

\tilde{h}

-regularity of ∞).

III. THE MAIN RESULTS

We now reformulate the main results of Theorems I.2 and I.3 in a broader context as a solution of the equivalent Problems 1–3.

Theorem III.1.

For arbitrary open set $Ω \subset R_{+}^{N + 1}$ the following conditions are equivalent:

$O$ is h-regular (or h-irregular).
Singular Parabolic Dirichlet Problem (2), and equivalently h-PDP has a unique (or infinitely many) solution(s).
Singular Parabolic Dirichlet Problem (2), and equivalently h-PDP is regular (or irregular) at $O$ ⁠.

Theorem III.2.

Let $Ω \subset R_{+}^{N + 1}$ be an open set satisfying (13) and (14). Then $O$ is h-regular if and only if the integral (15) diverges.

Similarly in the context of the $\tilde{h}$ -PDP the main results read:

Theorem III.3.

For arbitrary open set $\tilde{Ω} \subset R_{-}^{N + 1}$ the following conditions are equivalent:

∞ is $\tilde{h}$ -regular (or $\tilde{h}$ -irregular).
Singular Parabolic Dirichlet Problem (7), and equivalently $\tilde{h}$ -PDP has a unique (or infinitely many) solution(s).
Singular Parabolic Dirichlet Problem (7), and equivalently $\tilde{h}$ -PDP is regular (or irregular) at ∞.

Theorem III.4.

Let $\tilde{Ω} \subset R_{-}^{N + 1}$ be an open set satisfying (19) and (20). Then ∞ is $\tilde{h}$ -regular if and only if the integral (21) diverges.

A. Historical comments

The major problem in the Analysis of PDEs is understanding the nature of singularities of solutions to the PDEs reflecting the natural phenomena. It would be convenient to make some remarks on the analysis of singularities for the Laplace and heat equations, as well as more general second-order elliptic and parabolic PDEs. The solvability, in some generalized sense, of the classical DP in a bounded open set $E \subset R^{N}$ ⁠, with prescribed data on ∂E, is realized within the class of resolutive boundary functions, identified by Perron’s method and its Wiener^6,7 and Brelot⁸ refinements. Such a method is referred to as the PWB method, and the corresponding solutions are PWB solutions. Paralleling the theory of PWB solutions, the DP for the heat equation in an arbitrary open set is solvable within the class of resolutive boundary functions. We refer to Refs. 4 and 5 for an account of the theory. Wiener, in his pioneering works,^6,7 proved a necessary and sufficient condition for the finite boundary point x_o ∈ ∂E to be regular in terms of the “thinness” of the complementary set in the neighborhood of x_o. If the boundary of the domain is a graph in a neighborhood of x₀, the Wiener criterion is entirely geometrical. A key advance made in Wiener’s work was an introduction of the concept of capacity - sub-additive set function dictated by the Laplacian for the accurate measuring of the thinness of the complementary set in the neighborhood of x₀ for the boundary regularity of harmonic function. Formalized through the powerful Choquet capacitability theorem,⁹ the concept of capacity became a standard tool for the characterization of singularities for the elliptic and parabolic equations. The question of removability of isolated singularities for the linear second-order elliptic and parabolic PDEs was settled in Refs. 10–13. The Wiener criterion for the boundary continuity of harmonic functions became a canonical result driving the boundary regularity theory for the elliptic and parabolic PDEs. In 1935, Petrovsky proved a geometric necessary and sufficient condition for the regularity of the characteristic top boundary point for the heat equation in the domain of revolution² (see also Refs. 14). In the same paper, he also presented an elegant solution of the Kolmogorov problem (see Sec. I, Problem ${\tilde{A}}_{0}$ ⁠) for the special domain of revolution (see also Ref. 3). The results formed the so-called Kolmogorov–Petrovsky test for the asymptotic behavior of the standard Brownian path as t↓ 0 and t↑ + ∞, and opened a path for the deep connection between the regularity theory of elliptic and parabolic PDEs and asymptotic properties of the associated Markov processes.¹⁵ The geometric iterated logarithm test for the regularity of the boundary point for an arbitrary open set with respect to heat equation is proved in Ref. 16. Paralleling the Wiener regularity theory, Wiener’s criterion for the regularity of the finite boundary point for the heat equation was formulated in Ref. 17 along with the proof of the irregularity assertion. The problem was accomplished in Ref. 18, where the long-awaited regularity assertion was proved. As in its elliptic counterpart, the concept of heat capacity was a key concept to extend the Wiener regularity theory to the case of heat equation.⁵ However, the major technical difficulty in doing so was connected to the nature of singularities of the fundamental solution of the Laplace and heat equations. The former is an isolated singularity for the spherical level sets of the fundamental solution, while the latter is a non-isolated singularity point for the level sets of the fundamental solution of the heat equation. To complete the Wiener regularity result at finite boundary points for the heat equation, the major technical advance of paper¹⁸ was a proof of elegant boundary Harnack estimate near the non-isolated singularity point of the level sets of the fundamental solution to the backward heat equation. The result of¹⁸ was extended to the class of linear second-order divergence form parabolic PDEs with C¹-Dini continuous coefficients in Refs. 19 and 20.

In Ref. 21 it is proved that the Wiener test for the regularity of finite boundary points concerning second-order divergence form uniformly elliptic operator with bounded measurable coefficients coincides with the classical Wiener test for the boundary regularity of harmonic functions. The Wiener test for the regularity of finite boundary points for linear degenerate elliptic equations is proved in Ref. 22. The Wiener test for the regularity of finite boundary points for quasilinear elliptic equations was settled due to Refs. 23–25. Nonlinear potential theory was developed along the same lines as classical potential theory for the Laplace operator, for which we refer to Ref. 26.

To solve the DP in an unbounded open set, Brelot introduced the idea of compactifying $R^{N}$ into $R^{N} \cup {\infty}$ ⁠, where ∞ is the point at ∞ of $R^{N}$ ⁠.²⁷ The PWB-method is extended to the compactified framework, thus providing a powerful existence and uniqueness result for the DP in arbitrary open sets in the class of resolutive boundary functions. The new concept of regularity of ∞ was introduced in Ref. 28 for the classical DP, and in Ref. 1 for its parabolic counterpart. The DP with bounded Borel measurable boundary function has one and only one or infinitely many solutions without prescribing the boundary value at ∞. The point at ∞ is called a regular if there is a unique solution, and it is called irregular otherwise. Equivalently, in the measure-theoretical context, the new concept of regularity or irregularity of ∞ is introduced according to whether the harmonic measure of ∞ is null or positive. In Ref. 28 the Wiener criterion for the regularity of ∞ in the classical DP for the Laplace equation in an open set $E \subset R^{N}$ with N ≥ 3 is proved. In Ref. 29 it is proved that the Wiener criterion at ∞ for the linear second-order divergence form elliptic PDEs with bounded measurable coefficients coincides with the Wiener criterion at ∞ for the Laplacian operator. The Wiener criterion at ∞ for the heat equation is proved in Ref. 1. Remarkably, the Kolmogorov problem (see Sec. I, Problem ${\tilde{A}}_{0}$ ⁠) is a particular case of the problem of uniqueness of the bounded solution of the parabolic Dirichlet problem in arbitrary open set in $R^{N + 1}$ without prescribing the limit of the solution at ∞. Hence, the Wiener criterion at ∞ proved in Ref. 1 presents a full solution to the Kolmogorov problem.

The new Wiener criterion at ∞ for the elliptic and parabolic PDEs broadly extends the role of the Wiener regularity theory in mathematics. The Wiener test at ∞ arises as a global characterization of uniqueness in boundary value problems in arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criterion at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the Markov processes whose generator is a given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in a Dirichlet problem for a class of unbounded functions growing at a certain rate near the boundary point; a criterion for the removability of singularities and/or for unique continuation at the finite boundary point: let $E \subset R^{N}, N \geq 3$ be an open set, and x₀ ∈ E be a finite boundary point. Consider a singular Dirichlet problem for the linear second order uniformly elliptic PDE with bounded measurable coefficients in a class O(|x − x₀|^2−N) as x → x₀. In Ref. 29 it is proved that the Wiener test at x₀ is a necessary and sufficient condition for the unique solvability of the singular Dirichlet problem, and equivalently for the removability of the fundamental singularity at x₀. In a recent paper³⁰ an appropriate 2D analog of this result is established. Let $E \subset R^{2}$ be a Greenian open set, and x₀ ∈ ∂E be a boundary point (finite or ∞). Consider a singular Dirichlet problem for the linear second-order uniformly elliptic operator with bounded measurable coefficients in the class O(log |x − x₀|) if x₀ is finite, and in a class of functions with logarithmic growth, if x₀ = ∞. In Ref. 30 it is proved that the Wiener criterion at x₀ is a necessary and sufficient condition for the unique solvability of the singular Dirichlet problem, and equivalently for the removability of the logarithmic singularity. Precisely, in Ref. 30 the concept of log-regularity (or log-irregularity) of the boundary point (finite or ∞) is introduced according as if log-harmonic measure of it is null or positive, and the removability of the logarithmic singularity is expressed in terms of the Wiener criterion for the log-regularity of x₀.

The goal of this paper is to establish a necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular PDP. In this paper, we prove the Kolmogorov–Petrovsky-type test. We address the proof of the Wiener-type criterion in the forthcoming paper.

IV. PRELIMINARY RESULTS

The equivalence of two problems formulated in

R_{+}^{N + 1}

and

R_{-}^{N + 1}

is a consequence of the Appell transformation. Consider a homeomorphism

A : R_{+}^{N + 1} \cup {O} \mapsto R_{-}^{N + 1} \cup {\infty}

with

\{\begin{cases} (x, t) \in R_{+}^{N + 1} \mapsto A (x, t) = (\frac{x}{2 t}, - \frac{1}{4 t}) \in R_{-}^{N + 1}; A (O) = \infty \\ (x, t) \in R_{-}^{N + 1} \mapsto A^{- 1} (x, t) = (- \frac{x}{2 t}, - \frac{1}{4 t}) \in R_{+}^{N + 1}; A^{- 1} (\infty) = O \end{cases}

(44)

Let

P (Ω)

be a class of parabolic functions in an open set Ω. Given open set

Ω \subset R_{+}^{N + 1}

⁠, the Appell transformation is a homeomorphism

A : P (Ω) \mapsto P (A Ω)

defined as

\{\begin{cases} Ω \subset R_{+}^{N + 1} : u \in P (Ω) \mapsto A u (z) = {(- \frac{π}{t})}^{\frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} u (A^{- 1} (z)) \in P (A Ω) \\ Ω \subset R_{∹}^{N + 1} v \in P (Ω) \mapsto A^{- 1} v (z) = F (z) v (A (z)) \in P (A^{- 1} Ω) \end{cases}

(45)

The claim follows from the following formula:

\{\begin{cases} H [A u (z)] = \frac{π^{N / 2}}{4} {(- t)}^{- \frac{N}{2} - 2} e^{- \frac{| x |^{2}}{4 t}} H [u (A^{- 1} (z)], z \in A Ω \subset R_{-}^{N + 1} \\ H [A^{- 1} v (z)] = \frac{1}{4 t^{2}} F (z) H [v (A (z)], z \in A^{- 1} Ω \subset R_{+}^{N + 1} . \end{cases}

(46)

In particular, the Appell transform of h is given by

\tilde{h} (x, t) = A h (x, t) = e^{⟨ x, γ ⟩ + | γ |^{2} t},

(47)

as it is defined in .

From the formula (46) it follows that The Appell transformation is a homeomorphism between $S (Ω) \cap C^{2} (Ω)$ and $S (A Ω) \cap C^{2} (A Ω)$ ⁠, where $S (Ω)$ denotes the class of all superparabolic functions in Ω. A simple approximation argument can be used to demonstrate that the hypothesis C²(Ω) can be removed.⁴

Appell transformation presents one-to-one mapping between the singular PDPs (2) and (7).

Lemma IV.1.

Function u is h-parabolic (or h-superparabolic) in open set $Ω \subset R_{+}^{N + 1}$ if and only if u(A⁻¹(x, t)) is $\tilde{h}$ -parabolic (or $\tilde{h}$ -superparabolic) in $A Ω \subset R_{-}^{N - 1}$ ⁠.
${}^{h}H_{f}^{Ω}$ is a solution of the h-parabolic Dirichlet problem in $Ω \subset R_{+}^{N + 1}$ if and only if ${}^{h}H_{f}^{Ω} (A^{- 1} (\cdot))$ is a solution of the $\tilde{h}$ -parabolic Dirichlet problem in $A Ω \subset R_{-}^{N + 1}$ with boundary function f(A⁻¹), i.e.
${}^{h}H_{f}^{Ω} (A^{- 1} (z)) = {}^{\tilde{h}}H_{f (A^{- 1})}^{A Ω} (z), z = (x, t) \in A Ω .$
(48)
$O$ is h-regular for $Ω \subset R_{+}^{N + 1}$ if and only if ∞ is $\tilde{h}$ -regular for $A Ω \subset R_{-}^{N + 1}$ ⁠.
$H_{g}^{Ω}$ is a solution of the singular PDP (2) if and only if its Appell transform is a solution of the singular PDP (7) in AΩ with boundary function Ag(A⁻¹), i.e.
$A H_{g}^{Ω} (z) = H_{A g (A^{- 1})}^{A Ω} (z), z \in A Ω$
(49)
$A^{- 1} H_{\tilde{g}}^{\tilde{Ω}} (z) = H_{A^{- 1} \tilde{g} (A)}^{A^{- 1} \tilde{Ω}} (z), z \in A^{- 1} \tilde{Ω}$
(50)
Problems $A_{γ} |_{Ω}$ and ${\tilde{A}}_{γ} |_{A Ω}$ are equivalent, i.e. u_∗ ≡ 0 if and only if ${\tilde{u}}_{*} \equiv 0$ ⁠.

Proof.

Let $Ω \subset R_{+}^{N + 1}$ be an open set, and u be h-parabolic function on Ω, i.e.
$u (x, t) = \frac{v (x, t)}{h (x, t)}, (x, t) \in Ω,$
where v is a parabolic function in Ω. Considering the Appel transform of v = uh we have
$\begin{gathered} A (u h) (x, t) = {(- \frac{π}{t})}^{\frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} u (A^{- 1} (x, t)) h (A^{- 1} (x, t)) \\ = e^{\frac{| x + 2 γ t |^{2}}{4 t} - \frac{| x |^{2}}{4 t}} u (A^{- 1} (x, t)) = \tilde{h} (x, t) u (A^{- 1} (x, t)), (x, t) \in A Ω \subset R_{-}^{N + 1}, \end{gathered}$
which implies that u(A⁻¹(x, t)) is $\tilde{h}$ -parabolic function in AΩ. On the other side, let $Ω \subset R_{-}^{N + 1}$ be an open set, and u be $\tilde{h}$ -parabolic function on Ω, i.e.
$u (x, t) = \frac{v (x, t)}{\tilde{h} (x, t)}, (x, t) \in Ω,$
where v is a parabolic function in Ω. Considering the inverse Appel transform of $v = u \tilde{h}$ we have
$\begin{aligned} A^{- 1} (u \tilde{h}) (x, t) = F (x, t) u (A (x, t)) \tilde{h} (A (x, t)) \\ = {(4 π t)}^{- \frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} e^{\frac{2 ⟨ x, γ ⟩ - | γ |^{2}}{4 t}} u (A (x, t)) \\ = h (x, t) u (A (x, t)), (x, t) \in A^{- 1} Ω \subset R_{+}^{N + 1}, \end{aligned}$
which implies that u(A(x, t)) is h-parabolic function in A⁻¹Ω. The presented proof applies to smooth superparabolic functions without any changes. Using the standard smoothing, the proof is extended to h- and $\tilde{h}$ -superparabolic functions as well.
According to Claim (1) ${}^{h}H_{f}^{Ω} (A^{- 1} (x, t))$ is $\tilde{h}$ -parabolic in AΩ, and we only need to verify that the relations (23) and (24) corresponding to $\tilde{h}$ -parabolic Dirichlet problem are satisfied. For arbitrary z = (x, t) ∈ AΩ, we have
$\begin{gathered} {}^{h}{\bar{H}}_{f}^{Ω} (A^{- 1} (z)) \equiv \inf {u \in S_{h} (Ω) : \underset{\begin{array}{c} A^{- 1} (z) \to w \\ A^{- 1} (z) \in Ω \end{array}}{lim inf} u (A^{- 1} (z)) \geq f (w) \forall w \in \partial Ω} \equiv \\ \inf {u \in S_{\tilde{h}} (A Ω) : \underset{\begin{array}{c} z \to w \\ z \in A Ω \end{array}}{lim inf} u (z) \geq f (A^{- 1} (w)) \forall w \in \partial A Ω} \equiv {}^{\tilde{h}}{\bar{H}}_{f (A^{- 1})}^{A Ω} (z), \end{gathered}$
and
$\begin{gathered} {}^{h}{\underset{̲}{H}}_{f}^{Ω} (A^{- 1} (z)) \equiv \sup {u \in - S_{h} (Ω) : \underset{\begin{array}{c} A^{- 1} (z) \to w \\ A^{- 1} (z) \in Ω \end{array}}{lim sup} u (A^{- 1} (z)) \leq f (w) \forall w \in \partial Ω} \equiv \\ \sup {u \in - S_{\tilde{h}} (A Ω) : \underset{\begin{array}{c} z \to w \\ z \in A Ω \end{array}}{lim sup} u (z) \leq f (A^{- 1} (w)) \forall w \in \partial A Ω} \equiv {}^{\tilde{h}}{\underset{̲}{H}}_{f (A^{- 1})}^{A Ω} (z), \end{gathered}$
which implies (48).
If $f = 1_{{O}}$ ⁠, (48) implies
${}^{h}H_{1_{{O}}}^{Ω} (A^{- 1} (z)) = {}^{\tilde{h}}H_{1_{{\infty}}}^{A Ω} (z), z = (x, t) \in A Ω,$
(51)
which proves the claim.
By using (25) and (48) we have
$\begin{align} A H_{g}^{Ω} (z) = A [h {}^{h}H_{f}^{Ω}] (z) = {(- \frac{π}{t})}^{\frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} h (A^{- 1} (z)) {}^{h}H_{f}^{Ω} (A^{- 1} (z)) \\ = \tilde{h} (z) {}^{\tilde{h}}H_{f (A^{- 1})}^{A Ω} (z) = H_{\tilde{h} f (A^{- 1})}^{A Ω} (z) = H_{A g (A^{- 1})}^{A Ω} (z), z \in A Ω \end{align}$
(52)
since
$\begin{align} A g (A^{- 1} (z)) = {(- \frac{π}{t})}^{\frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} g (A^{- 1} (z)) \\ = {(- \frac{π}{t})}^{\frac{N}{2}} e^{- \frac{| x |^{2}}{4 t}} h (A^{- 1} (z)) f (A^{- 1} (z)) = \tilde{h} (z) f (A^{- 1} (z)), z \in A Ω . \end{align}$
(53)
Proof of the symmetric relation (50) is similar.
The claim is a direct consequence of (34) and (51).□

The next lemma expresses the fact that the property of h-regularity of the singularity point is local and order-preserving.

Lemma IV.2.

If $Ω_{1} \subset Ω_{2} \subset R_{+}^{N + 1}$ , then
- ${}^{h}H_{1_{{O}}}^{Ω_{2}} \equiv 0 \Rightarrow {}^{h}H_{1_{{O}}}^{Ω_{1}} \equiv 0$ ;
- ${}^{h}H_{1_{{O}}}^{Ω_{1}} ≢ 0 \Rightarrow {}^{h}H_{1_{{O}}}^{Ω_{2}} ≢ 0$ ;
Let $Ω \subset R_{+}^{N + 1}$ , and Ω_δ ≔ Ω ∩ {t<δ}, δ>0. Then ${}^{h}H_{1_{{O}}}^{Ω} \equiv 0$ if and only if ${}^{h}H_{1_{{O}}}^{Ω_{δ}} \equiv 0$ for some (and equivalently for all) δ > 0.
If ${\tilde{Ω}}_{1} \subset {\tilde{Ω}}_{2} \subset R_{-}^{N + 1}$ , then
- ${}^{\tilde{h}}H_{1_{{\infty}}}^{{\tilde{Ω}}_{2}} \equiv 0 \Rightarrow {}^{\tilde{h}}H_{1_{{\infty}}}^{{\tilde{Ω}}_{1}} \equiv 0$ ;
- ${}^{\tilde{h}}H_{1_{{\infty}}}^{{\tilde{Ω}}_{1}} ≢ 0 \Rightarrow {}^{\tilde{h}}H_{1_{{\infty}}}^{{\tilde{Ω}}_{2}} ≢ 0$ ;
Let $\tilde{Ω} \subset R_{-}^{N + 1}$ , and ${\tilde{Ω}}_{δ} ≔ \tilde{Ω} \cap {t < - δ}, δ > 0$ . Then ${}^{h}H_{1_{{\infty}}}^{\tilde{Ω}} \equiv 0$ if and only if ${}^{h}H_{1_{{\infty}}}^{{\tilde{Ω}}_{δ}} \equiv 0$ for some (and equivalently for all) δ > 0.
${}^{\tilde{h}}H_{1_{{\infty}}}^{D_{δ}} \equiv 0$ for $D_{δ} = {x \in R^{N}, t > - δ}, δ > 0$ ⁠.

Proof.

It is easy to see that ${}^{h}H_{1_{{O}}}^{Ω_{2}} = {}^{h}H_{g}^{Ω_{1}}$ on Ω₁, where

g = \{\begin{matrix} 1_{\{O\}}, on \partial Ω_{1} \cap \partial Ω_{2} \\ {}^{h}H_{1_{\{O\}}}^{Ω_{2}}, on \partial Ω_{1} \cap Ω_{2} . \end{matrix}

(54)

Since Perron’s solution is order-preserving, it follows that

{}^{h}H_{1_{{O}}}^{Ω_{1}} \leq {}^{h}H_{g}^{Ω_{1}} \leq {}^{h}H_{1_{{O}}}^{Ω_{2}}, on Ω_{1},

(55)

which implies the claims (1a) and (1b).

The “only if” claim is trivial. To demonstrate the “if” claim, note that since ∂Ω_δ ∩ {t = δ} is a parabolic measure null set for Ω_δ, we have

{}^{h}H_{1_{{O}}}^{Ω} |_{Ω_{δ}} = {}^{h}H_{1_{{O}}}^{Ω_{δ}} \equiv 0 .

Therefore, by the maximum principle we have

{}^{h}H_{1_{{O}}}^{Ω} |_{Ω \ {\bar{Ω}}_{δ}} \equiv {}^{h}H_{0}^{Ω \ {\bar{Ω}}_{δ}} \equiv 0,

which proves the claim .

The proof of (3) and (4) is identical to the proof of (1) and (2). The claim (5) is a consequence of the uniqueness result for the Cauchy problem.³¹

V. PROOFS OF MAIN RESULTS

A. Proof of theorem I.2

We assume that Ω satisfies and . Assume that the integral converges. We aim to demonstrate that u_∗ > 0, or equivalently,

{O}

is h-irregular. Without loss of generality, we can assume that ρ ∈ C¹(0, δ). Indeed, otherwise we can select the function ρ₁ ∈ C¹(0, δ) which satisfy

\frac{ρ (t)}{2} < ρ_{1} (t) < ρ (t), 0 < t < δ,

(56)

and consider the domain

Ω_{1} = {(x, t) : | x - γ |^{2} < 4 t \log ρ_{1} (t), 0 < t < δ} .

(57)

From it follows that the integral is convergent for ρ₁, and Ω₁ ⊂ Ω ∩ {0 < t < δ}. Therefore, h-irregularity of

{O}

for Ω₁ would imply so for Ω.

Consider a function

u (x, t) = 1 - ρ^{- 1} (t) e^{\frac{| x - γ |^{2}}{4 t}}

which is positive in Ω for 0 < δ ≪ 1, vanishes on ∂Ω ∩ {|x − γ|² = 4t log ρ(t)}, and satisfies

\lim_{t ↓ 0} u (γ, t) = \underset{(x, t) \to O, (x, t) \in Ω}{lim sup} u (x, t) = 1 .

(58)

For a function v = uh we have

\begin{align} H v = h [\frac{ρ^{'} (t)}{ρ^{2} (t)} e^{\frac{| x - γ |^{2}}{4 t}} + \frac{N}{2} \frac{1}{t ρ (t)} e^{\frac{| x - γ |^{2}}{4 t}} + \frac{1}{ρ (t)} \frac{| x - γ |^{2}}{2 t^{2}} e^{\frac{| x - γ |^{2}}{4 t}}] \\ - {(4 π t)}^{- \frac{N}{2}} \frac{1}{ρ (t)} \frac{| x - γ |^{2}}{2 t^{2}} = \frac{ρ^{'} (t)}{ρ^{2} (t)} {(4 π t)}^{- \frac{N}{2}} + \frac{N}{2} \frac{1}{t ρ (t)} {(4 π t)}^{- \frac{N}{2}} . \end{align}

(59)

Now we construct a function w with the following properties:

H w = - \frac{N}{2} \frac{1}{t ρ (t)} {(4 π t)}^{- \frac{N}{2}}, w (x, t) < 0, in Ω_{n},

(60)

\frac{w (γ, t)}{h (γ, t)} \geq - \frac{1}{2}, for n^{- 1} \leq t \leq T,

(61)

for some fixed 0 < T < δ and for all sufficiently large n. From and it follows that the function

\tilde{u} = \frac{v + w}{h}

is h-subparabolic, and by the maximum principle we have

u_{n} (x, t) \geq \tilde{u} (x, t) in Ω_{n}

(62)

If we select the fixed value T sufficiently small, it follows that

\tilde{u} (γ, T) > 1 / 3

⁠. Therefore, within Ω_T = Ω ∩ {t > T},

\tilde{u}

is greater than the function which is h-parabolic in Ω_T, takes the value 1/4 in {(x, t): |x − γ| ≤ ϵ, t = T} for some 0 < ϵ ≪ 1, and vanishes on the rest of the parabolic boundary of Ω_T. Hence, we have a positive lower bound for u_n in Ω_T which is independent of n. Obviously, the same lower bound holds for the limit function u_∗, that is to say the h-parabolic measure of

O

would be positive. Thus, to complete the proof we need to construct the function w with the properties and .

As a function w we select a particular solution of the equation from :

w (x, t) = - \frac{N}{2} \int_{n^{- 1}}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| x - ξ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{{(4 π τ)}^{- \frac{N}{2}}}{τ ρ (τ)} d ξ d τ

(63)

We only need to check that is satisfied. We have

|\frac{w (γ, t)}{h (γ, t)}| = \frac{N}{2} \int_{n^{- 1}}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ

We split the integral

\int_{n^{- 1}}^{t} = \int_{t / 2}^{t} + \int_{n^{- 1}}^{t / 2}

⁠, and estimate the first one as follows:

\begin{align} \frac{N}{2} \int_{t / 2}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ \leq \\ N 2^{\frac{N}{2} - 1} \int_{t / 2}^{t} \frac{1}{τ ρ (τ)} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} d ξ d τ \leq \\ N 2^{\frac{N}{2} - 1} π^{- \frac{N}{2}} \int_{R^{N}} e^{- | ξ |^{2}} d ξ \int_{t / 2}^{t} \frac{d τ}{τ ρ (τ)} = N 2^{\frac{N}{2} - 1} \int_{t / 2}^{t} \frac{d τ}{τ ρ (τ)} . \end{align}

(64)

From the convergence of the integral it follows that the right-hand side of converges to zero as t↓ 0. We then have

\begin{align} \frac{N}{2} \int_{n^{- 1}}^{t / 2} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ \leq \\ \frac{N}{2} \int_{n^{- 1}}^{t / 2} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} d ξ d τ \leq \\ ω_{N} \frac{N}{2} \int_{n^{- 1}}^{t / 2} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} {(2 π t)}^{- \frac{N}{2}} {(4 τ \log ρ (τ))}^{\frac{N}{2}} d τ \leq \\ ω_{N} \frac{N}{2} {(\frac{2}{π})}^{\frac{N}{2}} \int_{n^{- 1}}^{t / 2} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ, \end{align}

(65)

where ω_N is a volume of the unit ball in

R^{N}

⁠. From the convergence of the integral it follows that for some fixed value of T, is satisfied for all sufficiently large n. This completes the proof of the h-irregularity of the singularity point

O

⁠.

Let us prove the h-regularity of the singularity point

O

by assuming that the integral diverges. Without loss of generality, we assume that ρ satisfies the additional conditions

ρ \in C^{1} (0, δ),

(66)

\frac{t ρ^{'} (t)}{ρ (t)} \geq - L > - \frac{N}{2}

(67)

ρ (t) > | \log t | for 0 < t < δ .

(68)

The proof of the “if” statement is based on the construction of the family of h-superparabolic functions

{\tilde{u}}_{n}

with the following properties:

${\tilde{u}}_{n} (x, t) \geq 0$ in Ω_n;
$| 1 - {\tilde{u}}_{n} (x, n^{- 1}) | < \frac{1}{2}$ ⁠;
∀ϵ > 0 there exists a number T < δ such that ∀t₀ < T and for arbitrary sufficiently large n we have ${\tilde{u}}_{n} (x, t_{0}) < ϵ$ ⁠.

Indeed, the existence of such a family implies that $u_{n} (x, t_{0}) \leq 2 {\tilde{u}}_{n} (x, t_{0}) < 2 ϵ$ ⁠, for all large n. passing to the limit n↑ + ∞ it follows that u_∗(x, t₀) ≤ 2ϵ. From the maximum principle, it follows that u_∗(x, t) ≤ 2ϵ for all t > t₀. Since ϵ > 0 is arbitrary, the assertion of the theorem follows.

To construct such a family

{{\tilde{u}}_{n}}

⁠, we need a more precise asymptotic evaluation of

\frac{w}{h}

in Ω as t↓ 0. We have

\frac{w (γ, t)}{h (γ, t)} = - \frac{N}{2} \int_{n^{- 1}}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ .

(69)

We split the integral

\int_{n^{- 1}}^{t} = \int_{n^{- 1}}^{t μ (t)} + \int_{t μ (t)}^{t} ≕ I + J

⁠, where μ(t) = k log⁻² ρ(t) and k > 0 is a small number at our disposal. Next, we find the asymptotics of I as t↓ 0, and prove that it provides a dominating term for the asymptotic behavior of w/h. Since μ(t) → 0 as t↓ 0, we have tμ(t) ≪ t/2, and t − τ > t/2 for n⁻¹ ≤ τ ≤ tμ(t), and t sufficiently small. Therefore, we have

|- \frac{| ξ - γ |^{2}}{4 (t - τ)}| < \frac{| ξ - γ |^{2}}{2 t} < \frac{2 τ \log ρ (τ)}{t} < 2 μ (t) \log ρ (t μ (t)),

where the last inequality follows from the monotonicity of τ log ρ(τ). Indeed, from the assumption it follows that

{(τ \log ρ (τ))}^{'} = \log ρ (τ) + \frac{τ ρ^{'} (τ)}{ρ (τ)} > 0 as τ ↓ 0 .

(70)

From it follows that

\int_{t μ (t)}^{t} \frac{ρ^{'} (τ)}{ρ (τ)} d τ = \log ρ (t) - \log ρ (t μ (t)) \geq - \int_{t μ (t)}^{t} \frac{L}{τ} d τ = L \log μ (t),

which implies that

0 > 1 - \frac{\log ρ (t μ (t))}{\log ρ (t)} \geq L \frac{\log μ (t)}{\log ρ (t)} = L \frac{\log k - \log_{2} ρ (t)}{\log ρ (t)} ↑ 0,

and therefore,

\lim_{t ↓ 0} \frac{\log ρ (t μ (t))}{\log ρ (t)} = 1 .

(71)

Hence, we have

|- \frac{| ξ - γ |^{2}}{4 (t - τ)}| < \frac{3 k}{\log ρ (t)},

for all sufficiently small t. That is to say, ∀ ϵ > 0 we can find T > 0 such that ∀t < T

\exp (- \frac{| ξ - γ |^{2}}{4 (t - τ)}) > 1 - ϵ,

and therefore, we have

\begin{gathered} (1 - ϵ) \frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t μ (t)} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} {(\frac{t}{t - τ})}^{\frac{N}{2}} d τ \leq | I | \\ \leq \frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t μ (t)} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} {(\frac{t}{t - τ})}^{\frac{N}{2}} d τ \end{gathered}

where ω_N is the volume of the unit ball. Since n⁻¹ ≤ τ ≤ tμ(t), we have

1 \leq \frac{t}{t - τ} = 1 + \frac{τ}{t - τ} \leq 1 + \frac{μ (t)}{1 - μ (t)} ↓ 1, as t ↓ 0 .

Therefore, ∃ T > 0 such that ∀t < T, and for all sufficiently large n we have

(1 - ϵ) \frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t μ (t)} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ \leq | I | \leq (1 + ϵ) \frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t μ (t)} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ

Hence, the following asymptotic relation is proved

\lim_{t ↓ 0} \lim_{n ↑ + \infty} \frac{I}{\frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t μ (t)} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ} = 1 .

Since is divergent, it follows that

\lim_{t ↓ 0} \lim_{n ↑ + \infty} \frac{\frac{w (γ, t)}{h (γ, t)}}{- \frac{N ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ} = 1,

(72)

provided that the integrals J and

I_{1} = \int_{t μ (t)}^{t} \frac{| \log ρ (τ) |^{\frac{N}{2}}}{τ ρ (τ)} d τ

remain bounded as t↓ 0. We split the integral

\int_{t μ (t)}^{t} = \int_{t μ (t)}^{θ t} + \int_{θ t}^{t} ≕ I_{2} + I_{3}

⁠, with 0 < θ < 1 to be selected. For sufficiently small t we have

I_{3} < \int_{θ t}^{t} \frac{d τ}{τ} = \log \frac{1}{θ}, I_{2} < \int_{t μ (t)}^{θ t} \frac{d τ}{τ ρ^{\frac{1}{2}} (τ)} ≕ I_{4},

and we still need to demonstrate the boundedness of I₄ as t↓ 0. This will be proved below while proving the boundedness of the integrals I₅ and I₆.

Next, we estimate

\frac{w}{h}

inside Ω_n for small t. As before, we split the time integral as

\int_{n^{- 1}}^{t} = \int_{θ t}^{t} + \int_{t μ (t)}^{θ t} + \int_{n^{- 1}}^{t μ (t)} ≔ I_{5} + I_{6} + I_{7}

⁠. To estimate I₅, we use the identity

\frac{| x - ξ |^{2}}{4 (t - τ)} - \frac{| x - γ |^{2}}{4 t} = \frac{| t (ξ - γ) - τ (x - γ) |^{2}}{4 t τ (t - τ)} - \frac{| γ - ξ |^{2}}{4 τ},

and derive

\begin{gathered} | I_{5} | = \frac{N}{2} \int_{θ t}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{e^{- \frac{| t (ξ - γ) - τ (x - γ) |^{2}}{4 t τ (t - τ)}}}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} e^{\frac{| ξ - γ |^{2}}{4 τ}} d ξ d τ \\ = \frac{N}{2} \int_{θ t}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{e^{- \frac{{|\sqrt{\frac{t}{τ}} (ξ - γ) - \sqrt{\frac{τ}{t}} (x - γ)|}^{2}}{4 (t - τ)}}}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} e^{\frac{| ξ - γ |^{2}}{4 τ}} d ξ d τ \\ \leq \frac{N}{2} \int_{θ t}^{t} \frac{1}{τ} {(\frac{τ}{t})}^{- \frac{N}{2}} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{e^{- \frac{{|\sqrt{\frac{t}{τ}} (ξ - γ) - \sqrt{\frac{τ}{t}} (x - γ)|}^{2}}{4 (t - τ)}}}{{(4 π (t - τ))}^{\frac{N}{2}}} d ξ d τ \\ = \frac{N}{2} \int_{θ t}^{t} \frac{1}{τ} \int_{| ξ | \leq {(4 t \log ρ (τ))}^{\frac{1}{2}}} \frac{e^{- \frac{{|ξ - \sqrt{\frac{τ}{t}} (x - γ)|}^{2}}{4 (t - τ)}}}{{(4 π (t - τ))}^{\frac{N}{2}}} d ξ d τ \\ \leq \frac{N}{2} \int_{θ t}^{t} \frac{1}{τ} d τ π^{- \frac{N}{2}} \int_{R^{N}} e^{- | z |^{2}} d z = \frac{N}{2} \int_{θ t}^{t} \frac{d τ}{τ} = \frac{N}{2} \log \frac{1}{θ} . \end{gathered}

Let us estimate the integral

\begin{align} I_{6} = - \frac{N}{2} \int_{t μ (t)}^{θ t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| x - ξ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} e^{\frac{| x - γ |^{2}}{4 t}} d ξ d τ \\ = - \frac{N}{2} \int_{t μ (t)}^{θ t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (\frac{- τ | x - γ |^{2} - t | γ - ξ |^{2} - 2 t ⟨ x - γ, γ - ξ ⟩}{4 t (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ . \end{align}

(73)

Assuming

θ < \frac{1}{2}

⁠, from τ < θt it follows that t − τ > t/2. Therefore, we have

\begin{aligned} \frac{- τ | x - γ |^{2} - t | γ - ξ |^{2} - 2 t 〈 x - γ, γ - ξ 〉}{4 t (t - τ)} < \frac{〈 γ - x, γ - ξ 〉}{2 (t - τ)} < \frac{| x - γ ‖ γ - ξ |}{t} \\ < \frac{{(4 t \log ρ (t))}^{\frac{1}{2}} {(4 τ \log ρ (τ))}^{\frac{1}{2}}}{t} < 4 {(\frac{τ}{t})}^{\frac{1}{2}} {(\log ρ (t))}^{\frac{1}{2}} {(\log ρ (τ))}^{\frac{1}{2}} < 4 θ^{\frac{1}{2}} \log ρ (τ) . \end{aligned}

From we deduce

\begin{gathered} | I_{6} | \leq \frac{N ω_{N}}{2} \int_{t μ (t)}^{θ t} \frac{{(4 τ \log ρ (τ))}^{\frac{N}{2}} ρ^{4 θ^{\frac{1}{2}}} (τ)}{{(2 π t)}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d y d τ \\ = \frac{N ω_{N}}{2} {(\frac{2}{π})}^{\frac{N}{2}} \int_{t μ (t)}^{θ t} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ^{1 - 4 θ^{\frac{1}{2}}} (τ)} d τ . \end{gathered}

At this point, we are going to make a precise choice of the number θ. Since for arbitrary γ > 0

\frac{\log^{\frac{N}{2}} ρ (τ)}{ρ^{γ} (t)} \to 0 as t ↓ 0,

we can reduce the boundedness of |I₆| to the boundedness of |I₄| if we choose θ such that

1 - 4 θ^{\frac{1}{2}} > \frac{1}{2} or θ < \frac{1}{64} .

We fix the value

θ = \frac{1}{65}

⁠. Therefore, for sufficiently small t we have

| I_{6} | \leq \frac{N ω_{N}}{2} {(\frac{2}{π})}^{\frac{N}{2}} \int_{t μ (t)}^{\frac{1}{65} t} \frac{d τ}{τ ρ^{\frac{1}{2}} (τ)} d τ .

By using assumption we have

\begin{align} \int_{t μ (t)}^{\frac{t}{65}} \frac{d τ}{τ ρ^{\frac{1}{2}} (τ)} d τ \leq C \int_{t μ (t)}^{\frac{t}{65}} \frac{d τ}{τ | \log τ |^{\frac{1}{2}}} d τ = 2 [| \log t μ (t) |^{\frac{1}{2}} - {|\log \frac{t}{65}|}^{\frac{1}{2}}] \\ = 2 \frac{| \log t μ (t) | - |\log \frac{t}{65}|}{{|\log \frac{t}{65}|}^{\frac{1}{2}} + | \log t μ (t) |^{\frac{1}{2}}} = 2 \frac{| \log μ (t) | - \log 65}{{|\log \frac{t}{65}|}^{\frac{1}{2}} + | \log t μ (t) |^{\frac{1}{2}}} \\ 2 \frac{| \log μ (t) | - \log 65}{{|\log \frac{t}{65}|}^{\frac{1}{2}} [1 + {|\frac{\log t μ (t)}{\log t / 65}|}^{\frac{1}{2}}]} \leq \frac{| \log μ (t) |}{{|\log \frac{t}{65}|}^{\frac{1}{2}}} . \end{align}

(74)

By using a l’Hôpital’s rule and we have

\lim_{t \to 0} \frac{\log^{2} μ (t)}{- \log t / 65} = - 4 \lim_{t \to 0} \frac{2 \log \log ρ (t) - \log k}{\log ρ (t)} \frac{t ρ^{'} (t)}{ρ (t)} = 0,

(75)

and therefore, from we deduce that

\lim_{t \to 0} \int_{t μ (t)}^{\frac{t}{65}} \frac{d τ}{τ ρ^{\frac{1}{2}} (τ)} d τ = 0 .

(76)

Finally, we estimate the integral

\begin{gathered} I_{7} = - \frac{N}{2} \int_{n^{- 1}}^{t μ (t)} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| x - ξ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} e^{\frac{| x - γ |^{2}}{4 t}} d ξ d τ \\ = - \frac{N}{2} \int_{n^{- 1}}^{t μ (t)} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (\frac{- τ | x - γ |^{2} - t | γ - ξ |^{2} - 2 t ⟨ x - γ, γ - ξ ⟩}{4 t (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{1}{τ ρ (τ)} {(\frac{τ}{t})}^{- \frac{N}{2}} d ξ d τ . \end{gathered}

Our goal is to demonstrate that its asymptotics as t↓ 0 coincides with the asymptotics of the corresponding integral I in the expression of

\frac{w (γ, t)}{h (γ, t)}

⁠. To prove that we need to demonstrate that the term

\exp (\frac{- τ | x - γ |^{2} - 2 t ⟨ x - γ, γ - ξ ⟩}{4 t (t - τ)})

is close to 1 for small t. We have

|\frac{- τ | x - γ |^{2} - 2 t ⟨ x - γ, γ - ξ ⟩}{4 t (t - τ)}| \leq \frac{τ | x - γ |^{2}}{4 t (t - τ)} + \frac{| x - γ ‖ γ - ξ |}{2 (t - τ)}

Since

t - τ > \frac{t}{2}

⁠, we have

\frac{τ | x - γ |^{2}}{4 t (t - τ)} \leq \frac{2 τ \log ρ (t)}{t} \leq \frac{2 t μ (t) \log ρ (t)}{t} = \frac{2 k}{\log ρ (t)} \to 0, as t ↓ 0 .

Using and , we also deduce that for all sufficiently small t

\begin{gathered} \frac{| x - γ ‖ γ - ξ |}{2 (t - τ)} \leq \frac{| x - γ ‖ γ - ξ |}{t} \leq 4 {(\frac{τ \log ρ (τ) \log ρ (t)}{t})}^{\frac{1}{2}} \\ \leq 4 {(μ (t) \log ρ (t μ (t)) \log ρ (t))}^{\frac{1}{2}} = 4 k^{\frac{1}{2}} {(\frac{\log ρ (t μ (t))}{\log ρ (t)})}^{\frac{1}{2}} < 5 k^{\frac{1}{2}}, \end{gathered}

and the right-hand side will be sufficiently small if the parameter k at our disposal is chosen small enough.

Hence, we proved that for arbitrary ϵ > 0 there exists T > 0 such that for any fixed t < T and for all large n

|\frac{\frac{w (x, t)}{h (x, t)}}{\frac{w (γ, t)}{h (γ, t)}} - 1| < ϵ for (x, t) \in Ω_{n} .

Otherwise speaking,

\lim_{t ↓ 0} \lim_{n ↑ + \infty} \frac{\frac{w (x, t)}{h (x, t)}}{\frac{w (γ, t)}{h (γ, t)}} = 1, uniformly for all x with (x, t) \in Ω_{n} .

Consider a function

{\tilde{u}}_{n} (x, t) = \frac{\frac{v (x, t) + w_{1} (x, t)}{h (x, t)}}{\sup_{Ω_{n}} |\frac{w_{1} (x, t)}{h (x, t)}|} + 1,

where

w_{1} (x, t) = \frac{N - 2 L}{N} w (x, t) .

From , , and it follows that

H ({\tilde{u}}_{n} h) \geq 0

for small t. Therefore,

{\tilde{u}}_{n}

is h-superparabolic function for sufficiently small t. We can check that it satisfies the required conditions –.

We have $| \frac{v}{h} | < 1$ in Ω_n, w₁(x, n⁻¹) = 0, and $\frac{w_{1} (y, t)}{h (y, t)} \to - \infty$ as n↑ + ∞, t↓ 0. Therefore,

{\tilde{u}}_{n} (x, n^{- 1}) \to 1 as n ↑ + \infty uniformly for x .

(77)

We have

\lim_{t ↓ 0} \lim_{n ↑ + \infty} \frac{\frac{w_{1} (γ, t)}{h (γ, t)}}{- \frac{(N - 2 L) ω_{N}}{2 π^{\frac{N}{2}}} \int_{n^{- 1}}^{t} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ} = 1,

(78)

\lim_{n ↑ + \infty} \int_{n^{- 1}}^{t} \frac{\log^{\frac{N}{2}} ρ (τ)}{τ ρ (τ)} d τ = + \infty,

(79)

\lim_{t ↓ 0} \lim_{n ↑ + \infty} \frac{\frac{w_{1} (x, t)}{h (x, t)}}{\frac{w_{1} (γ, t)}{h (γ, t)}} = 1, uniformly for all x with (x, t) \in Ω_{n} .

(80)

From these conditions it follows that ∀ϵ > 0 there exists a number T < δ such that ∀0 < t₀ < T and for arbitrary sufficiently large n we have

{\tilde{u}}_{n} (x, t_{0}) < ϵ

⁠.

${\tilde{u}}_{n} (x, t) \geq 0$ in Ω_n, since $\frac{v}{h} \geq 0$ and $\frac{w_{1}}{h} \leq 0$ in Ω_n.

Hence we proved that the divergence of the integral (15) implies the h-regularity of $O$ for Ω provided that additional assumptions (66)–(68) are satisfied. Note that the assumptions (66)–(68) are satisfied for all functions in (16). Therefore, we completed the proof of h-regularity of $O$ and removability of the fundamental singularity for domains (18) with ϵ ≤ 0.

To complete the proof we only need to demonstrate that the assumptions – can be removed. Differentiability assumption can be removed as before with the only difference that we select ρ₁ ∈ C¹(0, δ) which satisfy

ρ (t) < ρ_{1} (t) < 2 ρ (t), 0 < t < δ,

(81)

and consider Ω₁ as in . From it follows that the integral is divergent for ρ₁, and Ω₁ contains Ω ∩ {t < δ}. Therefore, h-regularity of

O

for Ω₁ implies the h-regularity of

O

for Ω.

To remove assumption , assume on the contrary that there are arbitrarily small values of t such that ρ(t) ≤ |log t|. Consider a function

ρ_{1} (t) = \max (ρ (t); | \log t |)

Clearly, the h-regularity of

O

for Ω₁ implies the h-regularity of

O

for Ω. Hence, we only need to demonstrate that the integral is divergent for ρ₁. In view of our assumption we can choose the sequence {t_k} with the following properties:

\{\begin{cases} t_{k} > t_{k + 1} > \dots > 0, t_{k} ↓ 0 \\ ρ (t_{k}) = | \log t_{k} | \\ \frac{\log t_{k + 1}}{\log t_{k}} \geq 2 \end{cases}

(82)

Let us define a function

\tilde{ρ} (t) = | \log t_{k + 1} |, for t_{k + 1} \leq t < t_{k} .

We have

\tilde{ρ} (t) \geq ρ_{1} (t), 0 < t < t_{1} .

(83)

By the last relation of we have

\int_{t_{k + 1}}^{t_{k}} \frac{d t}{t \tilde{ρ} (t)} = 1 - \frac{\log t_{k}}{\log t_{k + 1}} \geq \frac{1}{2},

which implies that

\int_{0^{+}} \frac{d t}{t \tilde{ρ} (t)} = + \infty .

(84)

From and it follows that

\int_{0^{+}} \frac{d t}{t ρ_{1} (t)} = + \infty,

(85)

and therefore, the integral is also divergent for ρ₁. Hence, assumption is removable.

Next, we are going to demonstrate that the assumption is also removable. Assume that the given function ρ ∈ C¹(0, δ) has a divergent integral , but doesn’t satisfy the condition . Consider a one-parameter family of curves

ρ_{C} (t) = | \log C t |^{3}, C > 0, 0 < t < C^{- 1} .

(86)

Note that for any curve ρ_C the integral is convergent, and since

t^{- 1} \log^{\frac{N}{2}} t

is monotonically decreasing function for small t, the convergence of holds for any function satisfying ρ ≥ ρ_C near 0. Therefore, there are arbitrarily small values of t such that ρ(t) < ρ_C(t). Since ρ_C(C⁻¹) = 0, a graph of the given function ρ with divergent integral must intersect all curves ρ_C(t) with C ≥ δ⁻¹. For any point [ρ(t), t] on the positive quarter plane there exists a unique value

C = C (t) = t^{- 1} e^{- ρ^{\frac{1}{3}} (t)}

(87)

such that ρ_C(t) passes through the point [ρ(t), t]. Clearly, tC(t) → 0 as t → 0; and there exists a sequence t_n↓ 0 such that C(t_n)↑ + ∞ as n → +∞. We define a set

M = {t \in (0, δ] : C (t) = C_{1} (t)},

(88)

where

C_{1} (t) = \max_{t \leq τ \leq δ} C (τ)

⁠. Denote by

{\bar{M}}^{c}

the complement of

\bar{M}

⁠. Since

{\bar{M}}^{c}

is an open set, we have

{\bar{M}}^{c} = \cup_{n} (t_{2 n}, t_{2 n - 1}) .

(89)

From the definition of the set M it follows that C(t) satisfies the following properties:

\{\begin{cases} C (t) is a decreasing function for t \in \bar{M} \\ C (t_{2 n}) = C (t_{2 n - 1}) for (t_{2 n}, t_{2 n - 1}) \subset {\bar{M}}^{c} \\ C (t) ↑ + \infty as \bar{M} ∋ t ↓ 0 . \end{cases}

(90)

Indeed, if we take t′, t″ ∈ M with t′ > t″, then we have

C (t^{″}) = C_{1} (t^{″}) \geq C_{1} (t^{'}) = C (t^{'}) .

On the other hand, if

t^{'}, t^{″} \in \bar{M}

the same conclusion follows from the continuity of the function C(t).

To prove the second assertion, first note that since $t_{2 n}, t_{2 n - 1} \in \bar{M}$ ⁠, we have C₁(t_2n) = C(t_2n) and C₁(t_2n−1) = C(t_2n−1). Therefore, assuming that C(t_2n) ≠ C(t_2n−1) would imply that C₁(t_2n) > C₁(t_2n−1). Since C₁ is a continuous function for some ϵ ∈ (0, t_2n−1 − t_2n) we have C₁(t_2n−1) < C₁(t_2n + ϵ). Let C₁(t_2n + ϵ) = C(θ). Obviously, we must have θ ∈ [t_2n + ϵ, t_2n−1) and C₁(θ) = C(θ). However, this is a contradiction with the fact that $(t_{2 n}, t_{2 n - 1}) \subset {\bar{M}}^{c}$ ⁠, which proves the second assertion of (90).

To prove the third assertion of , recall that there is a sequence t_n↓ 0 with C(t_n)↑ + ∞. Assume that for some n we have

t_{n} \notin \bar{M}

⁠. Then there exists k_n such that

t_{n} \in (t_{k_{n}}, t_{k_{n} + 1}) \subset {\bar{M}}^{c}

⁠. According to the second assertion of we have

C (t_{n}) \leq C_{1} (t_{n}) \leq C_{1} (t_{k_{n}}) = C (t_{k_{n}}) = C (t_{k_{n} + 1})

Consider a sequence

{\tilde{t}}_{n} = \{\begin{cases} t_{n}, if t_{n} \in \bar{M}, \\ t_{k_{n}}, if t_{n} \in (t_{k_{n}}, t_{k_{n} + 1}) \subset {\bar{M}}^{c} . \end{cases}

(91)

Clearly, we have

{\tilde{t}}_{n} \in \bar{M}; {\tilde{t}}_{n} ↓ 0

and

C ({\tilde{t}}_{n}) ↑ + \infty

as n↑ + ∞. From the monotonicity of C(t) on

\bar{M}

we easily deduce the third assertion of .

Let us define now a new function ρ₁ with the following properties:

ρ₁(t) = ρ(t) for $t \in \bar{M}$ ⁠,
ρ₁(t) = |log(C(t_2n)t)|³ for $t \in (t_{2 n}, t_{2 n - 1}) \subset {\bar{M}}^{c}$ ⁠.

Note that equivalently we can define ρ₁ as

ρ_{1} (t) = | \log (C_{1} (t) t) |^{3}, 0 < t \leq δ .

(92)

In fact, function C(t) defined for the function ρ₁ via coincides with C₁(t), i.e.

C (t) = \max_{t \leq τ \leq δ} C (τ) .

The new function is continuous and satisfy ρ₁(t) ≥ ρ(t) everywhere, with ρ₁(t) ≠ ρ(t) on intervals (t_2n, t_2n−1). It can be easily seen that the function ρ satisfies the desired property on

\bar{M}

⁠. Indeed, for a function ρ_C we have

\frac{t ρ_{C}^{'} (t)}{ρ_{C} (t)} = \frac{- 3}{| \log C t |} \to 0, as C t \to 0 .

Since C(t) is monotonically decreasing on

\bar{M}

⁠, we have

| ρ^{'} (t) | \leq | ρ_{C}^{'} (t) | = |\frac{3 \log^{2} (C t)}{t}|, t \in \bar{M}

provided that C = C(t) is chosen as in . Therefore, we have

|\frac{t ρ^{'} (t)}{ρ (t)}| = |\frac{3}{\log (C (t) t)}|, t \in \bar{M} .

(93)

Since C(t)t → 0 as t↓ 0, the right-hand side is arbitrarily small for small t, and clearly ρ satisfies on

\bar{M}

⁠.

Our next goal is to demonstrate that

\int_{{\bar{M}}^{c}} \frac{\log^{\frac{N}{2}} ρ (t)}{t ρ (t)} d t < + \infty

(94)

Since ρ ≥ ρ₁ it is satisfactory to demonstrate that the integral is convergent for ρ₁. By using the property of ρ₁ we have

\int_{{\bar{M}}^{c}} \frac{\log^{\frac{N}{2}} ρ_{1} (t)}{t ρ_{1} (t)} d t = \sum_{n} \int_{t_{2 n}}^{t_{2 n - 1}} \frac{3^{\frac{N}{2}} \log^{\frac{N}{2}} | \log C_{n} t |}{t | \log C_{n} t |^{3}} d t,

(95)

where we use a notation C_n ≔ C(t_2n) ≡ C(t_2n−1). Since C_nt is arbitrarily small for t sufficiently small, we can use the following inequality for all large n:

\frac{3^{\frac{N}{2}} \log^{\frac{N}{2}} | \log C_{n} t |}{| \log C_{n} t |} \leq 1

(96)

Using , from we deduce

\int_{{\bar{M}}^{c}} \frac{\log^{\frac{N}{2}} ρ_{1} (t)}{t ρ_{1} (t)} d t \leq \sum_{n} \int_{t_{2 n}}^{t_{2 n - 1}} \frac{d t}{t | \log C_{n} t |^{2}} d t = \sum_{n} \frac{1}{| \log C_{n} t_{2 n - 1} |} - \frac{1}{| \log C_{n} t_{2 n} |}

(97)

Since

ρ (t_{2 n - 1}) = | \log C_{n} t_{2 n - 1} |^{3}, ρ (t_{2 n}) = | \log C_{n} t_{2 n} |^{3},

and both ρ(t) and |log t|³ are decreasing functions, we deduce that

C_{n} t_{2 n - 1} \geq C_{n} t_{2 n} \geq C_{n + 1} t_{2 n + 1}, n = 1, 2, \dots

(98)

Therefore we also have

\frac{1}{| \log C_{n} t_{2 n - 1} |} \geq \frac{1}{| \log C_{n} t_{2 n} |} \geq \frac{1}{| \log C_{n + 1} t_{2 n + 1} |}, n = 1, 2, \dots

(99)

Hence, the series is a telescoping series and therefore integrals and are convergent integrals. Since ρ₁ ≤ ρ on

{\bar{M}}^{c}

⁠, it follows that the integral is convergent for ρ. On the other side, since the integral is divergent it follows that

\int_{\bar{M}} \frac{\log^{\frac{N}{2}} ρ (t)}{t ρ (t)} d t = + \infty .

(100)

Now we pursue the identical proof given above under the assumptions – by replacing the function with the following one:

\tilde{w} (x, t) = - \frac{N}{2} \int_{n^{- 1}}^{t} \int_{| ξ - γ | \leq {(4 τ \log ρ (τ))}^{\frac{1}{2}}} \frac{\exp (- \frac{| x - ξ |^{2}}{4 (t - τ)})}{{(4 π (t - τ))}^{\frac{N}{2}}} \frac{{(4 π τ)}^{- \frac{N}{2}}}{τ \tilde{ρ} (τ)} d ξ d τ,

(101)

where the function

\tilde{ρ}

in the integrand is chosen with the following properties:

$\tilde{ρ} (t) = ρ (t)$ for $t \in \bar{M}$ ⁠;
In the complementary set ${\bar{M}}^{c}$ the function $\tilde{ρ}$ is chosen as sufficiently large continuous function with the property that all the estimations of the function w/h from (69) in the proof given above remain valid when the function ρ in the integrand is replaced with $\tilde{ρ}$ ⁠.
$\tilde{ρ} (t) \geq ρ (t), 0 < t \leq δ$ ⁠.

To estimate the function $\tilde{w} / h$ in Ω as t↓ 0, the time integral in (n⁻¹, t) is split into two parts over $\bar{M} \cap (n^{- 1}, t)$ and ${\bar{M}}^{c} \cap (n^{- 1}, t)$ ⁠. The estimation of the first one is identical to the presented proof, for the assumptions (66)–(68) are satisfied on $\bar{M}$ ⁠. Due to property (2) of the function $\tilde{ρ}$ the second integral remains bounded and accordingly does not affect the leading asymptotic of $\tilde{w} / h$ given via the divergent integral (100). Precisely, we establish (72) and (77)–(80), where the integral term $\int_{n^{- 1}}^{t}$ in expressions (78) and (79) is replaced with $\int_{\bar{M} \cap (n^{- 1}, t)}$ ⁠. This completes the proof of the h-regularity of $O$ without assumptions (66)–(68).□

Theorems I.3 follow from the Lemma IV.1 (iv) and the mapping (44).

B. Proof of theorems III.1–III.4

According to the Definitions II.4, II.5, and formulae (34) and(39), Theorem I.2 and I.3 are equivalent to the Theorems III.2 and III.4 respectively.

The equivalence (1) ⇔ (2) in Theorems III.1 and III.3 follows from the formulae (29), (31), (37), and (38).

The equivalence (2) ⇔ (3) in Theorem III.3 with γ = 0 is proved in Ref. 1 (see Lemma 2.3, p. 472). Applying Lemma IV.1 (4), the equivalence (2) ⇔ (3) in Theorem III.1 with γ = 0 follows. Applying the translation x ↦ x + γ, the equivalence of (2) ⇔ (3) in Theorem III.1 with γ ≠ 0 easily follows. Applying Lemma IV.1 (iv) again, the equivalence (2) ⇔ (3) in Theorem III.3 with γ ≠ 0 follows.□

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Ugur G. Abdulla: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Removability of the fundamental singularity for the heat equation and its consequences

I. PRELUDE

A. Overture: Kolmogorov–Petrovsky-type test for the removability of the fundamental singularity

II. FORMULATION OF PROBLEMS

A. Unique solvability of the singular parabolic Dirichlet problem

B. Characterization of the h-parabolic measure of singularity point

C. Boundary regularity in singular Dirichlet problem

III. THE MAIN RESULTS

A. Historical comments

IV. PRELIMINARY RESULTS

V. PROOFS OF MAIN RESULTS

A. Proof of theorem I.2

B. Proof of theorems III.1–III.4

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Removability of the fundamental singularity for the heat equation and its consequences

I. PRELUDE

A. Overture: Kolmogorov–Petrovsky-type test for the removability of the fundamental singularity

II. FORMULATION OF PROBLEMS

A. Unique solvability of the singular parabolic Dirichlet problem

B. Characterization of the h-parabolic measure of singularity point

C. Boundary regularity in singular Dirichlet problem

III. THE MAIN RESULTS

A. Historical comments

IV. PRELIMINARY RESULTS

V. PROOFS OF MAIN RESULTS

A. Proof of theorem I.2

B. Proof of theorems III.1–III.4

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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