We study the optimality of the remainder term in the two-term Weyl law for the Dirichlet Laplacian within the class of Lipschitz regular subsets of $Rd$. In particular, for the short-time asymptotics of the trace of the heat kernel, we prove that the error term cannot be made quantitatively better than little-*o* of the second term.

## I. INTRODUCTION AND MAIN RESULTS

Let −Δ_{Ω} denote the Dirichlet Laplace operator on an open set $\Omega \u2282Rd$, which is defined as a self-adjoint operator in *L*^{2}(Ω) through the quadratic form *u* ↦∫_{Ω}|∇*u*(*x*)|^{2} *dx* with form domain $H01(\Omega )$. If the measure of $\Omega \u2282Rd$ is finite, the spectrum of −Δ_{Ω} is discrete and consists of an infinite number of positive eigenvalues accumulating only at infinity. Here, the eigenvalues are denoted by

where each eigenvalue is repeated according to its multiplicity.

The study of the asymptotic behavior of *λ*_{k} as *k* → *∞* is a classical topic in spectral theory. The most fundamental result in this area is the following celebrated result going back to Weyl,^{1} which states that

Here and in what follows, *ω*_{d} denotes the volume of the *d*-dimensional unit ball. That (1) holds for any open set $\Omega \u2282Rd$ of finite measure was obtained in Ref. 2.

If the set Ω has certain geometric properties, a refined version of the asymptotic expansion (1) holds, namely,

Here and in what follows, $Hd\u22121(A)$ denotes the (*d* − 1)-dimensional Hausdorff measure of a set $A\u2282Rd$. This refinement of Weyl’s law was conjectured already by Weyl.^{3} A satisfactory answer remained elusive for several decades, but Ivrii^{4} proved the conjecture under the assumption that Ω is smooth and the measure of the periodic billiards in Ω is zero.

### A. Main results

In this paper, our focus is on the remainder term in (2) or rather the corresponding remainder term in certain averages of the counting function. The greater part of our analysis concerns the remainder term in the Abel-type average

that is, the short-time asymptotics of the trace of the heat kernel. The asymptotics (3) can be obtained from (2) by integration in *λ*. However, it is not possible to reverse this process and deduce (2) from (3) alone.

Intuitively, the averaging of the eigenvalues should have a regularizing effect on the asymptotics, and thus, one expects (3) to be valid under less restrictive geometric assumptions than those needed for (2). Under the weak assumption that the boundary of Ω is Lipschitz regular, the validity of (3) was proved by Brown.^{5} In the same paper, Brown remarked without proof that the error term $o(t)$ cannot be replaced by *o*(*t*^{1/2+ϵ}) for any *ϵ* > 0. The main theorem of this paper goes in the same direction as this remark and, in fact, contains the remark as a particular case. However, our result claims substantially more. While Brown’s remark concerns the impossibility of improving the error term on the algebraic scale, our result states that it is impossible to make any quantitative improvement whatsoever. Specifically, we prove the following theorem:

*Let*$g:R+\u2192R$

*be a non-negative function with*$limt\u21920+g(t)=0$

*. There exists an open, bounded, and connected set*$\Omega \u2282Rd$

*with Lipschitz regular boundary such that*

In addition to considering the trace of the heat kernel (3), we will consider the so-called Riesz means of order *γ* ≥ 0, which are defined by

where $x\xb1=12(|x|\xb1x)$. In particular, $Tr(\u2212\Delta \Omega \u2212\lambda )\u22120=#{\lambda k<\lambda}$. The quantities $Tr(\u2212\Delta \Omega \u2212\lambda )\u2212\gamma $ become more and more well-behaved as *γ* increases. Furthermore, by setting *γ* = *tλ* and normalizing appropriately, one obtains $Tr(et\Delta \Omega )$ in the limit *λ* → *∞*.

Again by integration in *λ*, one can deduce a two-term asymptotic formula for the Riesz means from (2), which reads

as *λ* → *∞*, where we abbreviate

As in the case of the trace of the heat kernel, the regularizing effect of averaging should help us to prove the validity of (5) in greater generality for larger *γ*. In a recent paper,^{6} it is proved that (5) is valid for *γ* ≥ 1 as soon as the boundary of Ω is Lipschitz regular. This implies Brown’s result,^{5} but not the other way around.

Our interest in the current topic was motivated by the question of sharpness for the result obtained in Ref. 6. As a corollary of Theorem 1.1, we obtain that the remainder *o*(*λ*^{γ+(d−1)/2}) in the asymptotic expansion in (5) cannot be improved.

*Let*$R:R+\u2192R$

*be a non-negative function with*$lim\lambda \u2192\u221eR(\lambda )=0$

*. There exists an open, bounded, and connected set*$\Omega \u2282Rd$

*with Lipschitz regular boundary such that for all*

*γ*≥ 0,

As Theorem 1.2 concerns the lim sup and not the limit, the result is somewhat weaker than what could be expected from Theorem 1.1. As such, it is reasonable that it should follow from the same principal ideas. However, we are unable to give a direct proof. The main advantage in working with the trace of the heat kernel in comparison to $Tr(\u2212\Delta \Omega \u2212\lambda )\u2212\gamma $ lies in that we can utilize pointwise estimates for the heat kernel. The corresponding estimates for $Tr(\u2212\Delta \Omega \u2212\lambda )\u2212\gamma $ are much more delicate. However, it is not very surprising that one can deduce Theorem 1.2 from Theorem 1.1. Indeed, by the identity

an asymptotic expansion for $Tr(\u2212\Delta \Omega \u2212\lambda )\u2212\gamma $ as *λ* → *∞* implies a corresponding expansion for $Tr(et\Delta \Omega )$ as *t* → 0^{+}. However, the use of (6) in our Proof of Theorem 1.2 leads to the lim sup instead of the limit.

### B. Additional remarks

In a direction similar to that of Theorems 1.2 and 1.1, it was shown by Lazutkin and Terman^{7} that the error term in (2) cannot be improved on the algebraic scale even among planar convex sets, which, in addition, satisfy the assumptions of Ivrii’s result. It is interesting to note that if one considers not the asymptotics of the counting function but of either $Tr(\u2212\Delta \Omega \u2212\lambda )\u2212\gamma $, with *γ* ≥ 1, or $Tr(et\Delta \Omega )$, one *can* improve the error term on the algebraic scale within convex sets (in any dimension). This is the case that can be deduced from the following uniform inequality proved by the authors in Ref. 6. There exists a constant *C* > 0 such that for any convex domain $\Omega \u2282Rd$ and all *λ* ≥ 0,

where *r*(Ω) denotes the inradius of Ω. The corresponding inequality for *γ* > 1 and $Tr(et\Delta \Omega )$ follows from (7) through integration in *λ*.

For $\Omega \u2282R2$, a piecewise smooth domain with finitely many corners, it is possible to refine the asymptotic expansions discussed by yet another term^{8–12} (see also Ref. 13 for a similar result in polyhedra in $Rd$). Let the boundary of Ω be given by the union of smooth curve segments *γ*_{j}, *j* = 1, …, *m*, parameterized by the arc length and ordered so that *γ*_{j} meets *γ*_{j+1} and *γ*_{m} meets *γ*_{1}. Let *α*_{j} ∈ (0, 2*π*) denote the interior angle formed at the point *γ*_{j} ∩ *γ*_{j+1}. Then, as *t* → 0^{+},

where *κ*(*s*) denotes the curvature. It is clear that this expansion cannot extend to the class of Lipschitz sets, as the third term need not be finite.

There has been interesting work on Weyl asymptotics in very irregular sets, specifically sets with fractal boundary. In particular, a lot of work has been directed toward the optimal order of the error term in (1) in this setting. In 1979, it was conjectured by Berry^{14,15} that if *∂*Ω has the Hausdorff dimension $d\u22121<dH\u2264d$, then

However, it was shown by Brossard and Carmona^{16} that as stated, the conjecture of Berry needs to be modified, and they suggested that the Hausdorff dimension of *∂*Ω should be replaced by the Minkowski dimension. Subsequently, it was proved by Lapidus^{17} that if *∂*Ω has the Minkowski dimension $dM$ and a finite $dM$-dimensional Minkowski content, then the error in the Weyl formula is $O(\lambda dM/2)$. We emphasize, however, that the sets considered here, while having a non-trivial structure on all scales such as fractals, are much more regular. Nevertheless, the crucial feature of the set that we construct in the proof of Theorem 1.1 and 1.2 is that its boundary has non-trivial structure on all microscopic scales.

## II. PROOF OF THEOREM 1.1

### A. Reduction to the two-dimensional case

In order to simplify the construction, we first show that it is sufficient to consider the two-dimensional case. Assume that Theorem 1.1 is known in the case *d* = 2. Let *g* be as in the statement of Theorem 1.1 and fix a Lipschitz regular open, bounded, and connected set $\Omega \u2282R2$, satisfying (4) with *g* replaced by $g\u0303(t)=max{g(t),t1/2}$. We claim that Ω′ = Ω × (0,1)^{d−2} satisfies (4). Clearly, Ω′ is open, bounded, connected, and Lipschitz regular.

By the product structure of Ω′,

Moreover,

It is not difficult to show that

In fact, an explicit computation based on the Poisson summation formula and the explicit formulas for the eigenfunctions and eigenvalues on the interval yields the following lemma:

*The Dirichlet heat kernel of*−Δ

_{(0,L)}

*evaluated on the diagonal is given by*

*L*= 1. Recall that

*λ*

_{k}=

*π*

^{2}

*k*

^{2}with the corresponding

*L*

^{2}-normalized eigenfunction $\phi k(x)=2sin(\pi kx)$. By the explicit formulas for the eigenfunctions and eigenvalues,

By integrating (11), one obtains

Thus, by the product structure of (0,1)^{n},

which is the claimed expansion (10).

so

where in the last step, we used the choice of Ω and the fact that by construction, $g\u0303(t)\u2265g(t)$ and $t1/2\u2264g\u0303(t)$. This proves our claim and consequently reduces the Proof of Theorem 1.1 to the two-dimensional case.

### B. Geometric construction

We construct a domain $\Omega \u2282R2$ as follows: The idea is to begin with the square *Q*_{0} = (0,3)^{2} to which we add triangular teeth to the top edge, all separated from each other and of the same shape but of different sizes and all away from the vertical edges of the large square (see Fig. 1).

Precisely, we consider

where

where the sequence ${lk}k\u22651$ is positive and non-increasing with *∑*_{k}*l*_{k} ≤ 1, and the *c*_{k} are chosen increasing and such that 1 ≤ *c*_{k} ≤ min{*c*_{k+1} − *l*_{k}, 2}. This ensures that the supports of the different copies of *H*_{0} are disjoint and at least a distance 1 away from the vertical edges of the square *Q*_{0}. For instance, the sequence given by *c*_{1} = 1 and $ck=1+\u2211j=1k\u22121lj$ for *k* > 1 satisfies all the requirements. Note that at all points *x*, where *H* is differentiable, we have *H*′(*x*) ∈ {0, 1, −1}. Thus, the set Ω defined by (12) is open, bounded, and connected and has Lipschitz regular boundary.

The idea is that the teeth at a scale much smaller than $t$ should not have an essential contribution to the trace of the heat kernel. Moreover, the decrease in the area in removing such teeth is small relative to the decrease in the length of the boundary. We emphasize that it is not the presence of corners that we are playing with to construct our counterexample. The effect that is essential for our construction is rather that the boundary has a non-trivial structure on all scales. Heuristically, the construction should go through if the triangular teeth were replaced by versions, where each of the three corners had been smoothed out. However, even though such a modification would make *H*_{0} smooth, the function *H* and the boundary of the set Ω would be remain merely Lipschitz due to all derivatives of order greater than 1 blowing up as *l*_{k} tends to zero.

Let Ω_{M} be the domain described similarly to Ω by (12), but with the function *H* replaced by *H*_{M},

That is, we remove all teeth after the *M*th one. We note that Ω_{M′} ⊂ Ω_{M} ⊂ Ω, if *M*′ < *M*, ∪_{M≥1}Ω_{M} = Ω, and by construction,

By monotonicity of Dirichlet eigenvalues under set inclusions,

where we define

Our goal is to choose ${lk}k\u22651$ and *M* and to bound *E*_{M}(*t*) in such a way that

Equivalently, by (14), we want to achieve

Before we are able to conclude our proof, we shall need to prove some auxiliary results. The first result that we shall need is a bound for *E*_{M}(*t*). This is the content of Proposition 3.1 proved in Sec. III, which states that

Inserting this bound into (16) yields

We claim that one can choose a sequence *l*_{k} and a decreasing function *M*(*t*) in such a manner that the quantity (17) tends to infinity as *t* → 0^{+}. Indeed, setting $\tau =t$ and defining *G*(*τ*) = *g*(*τ*^{2}), the existence of such a sequence follows from Corollary 4.2 proved in Sec. IV. Note that for any *l*_{k} and *M* provided by Corollary 4.2, one has necessarily $limt\u21920+M(t)=\u221e$ and therefore $(4|\Omega M|\u22121)t=o(Mt)$. Thus, for the *l*_{k} and *M* provided by Corollary 4.2, only the first term in (17) affects the limit. Contingent on us proving Proposition 3.1 and Corollary 4.2, this completes the Proof of Theorem 1.1.

## III. AN ESTIMATE FOR THE ERROR TERM

Recall that

with *H*_{M} given by (13), and *E*_{M}(*t*) is defined in (15). Our goal in this section is to prove the following proposition:

*Let*${lk}k\u22651$

*be a non-negative sequence with*

*∑*

_{k≥1}

*l*

_{k}≤ 1

*. For every*

*M*≥ 1

*and*

*t*> 0

*,*

*M*and

*t*in Proposition 3.1 appears to be order-sharp. Indeed, for each fixed

*M*, the three-term expansion (8) states that

Let $H\Omega (x,t)=et\Delta \Omega (x,x)$ denote the heat kernel of the Dirichlet Laplacian on Ω evaluated on the diagonal. For our Proof of Proposition 3.1, we need a pointwise lower bound for $H(0,L)2(x,t)$ that has the asymptotically correct behavior close to the boundary.

*For the heat kernel of Dirichlet Laplacian on the interval*(0,

*L*)

*,*

*m*≥ 1 and

*x*∈ (0,

*L*).□

*For the heat kernel of Dirichlet Laplacian on the square*(0,

*L*)

^{2},

*we have*

*where*

*x*= (

*x*

_{1},

*x*

_{2}) ∈ (0,

*L*)

^{2}

*and*

*d*(

*x*

_{j}) = min{

*x*

_{j},

*L*−

*x*

_{j}}

*.*

*x*

_{1},

*x*

_{2}≤

*L*/2. By Lemma 3.3 and the product structure of the heat kernel in (0,

*L*)

^{2}, we have

*A*

_{+}

*B*

_{+}= (

*A*+

*A*

_{−})(

*B*+

*B*

_{−}), we find

*L*−

*x*

_{j}≥

*L*/2 and, since $1\u2212exj2/t\u22650$,

We are now ready to prove Proposition 3.1.

Our proof is based on the fact that if *x* ∈Ω ⊂Ω′, then *H*_{Ω}(*x*, *t*) ≤ *H*_{Ω′}(*x*, *t*) for any *t* > 0.^{18} We will bound $Tr(et\Delta \Omega M)$ from below by bounding the heat kernel pointwise from below in terms of the heat kernel of appropriate squares contained in Ω_{M}.

*δ*,

*t*> 0,

We introduce some notation for different regions in Ω. Let *Q*_{k} be the square of side-length $2$ placed so that one corner matches the *k*th tooth. By construction, each tooth is at least a distance 1 away from the vertical sides of *Q*_{0} = (0,3)^{2}. Therefore, *Q*_{k} ⊂ Ω_{M} for each *k* < *M*. Let *M*_{k} be the triangular region corresponding to the *k*-th tooth, and let *L*_{k} denote the same region but mirrored across the boundary of *Q*_{0}. The set $Lk\u222aMk\xaf$ is a square with sidelength $lk/2$. Since $lk2<\u2211j\u22651lj2\u226412=22$, the set *L*_{k} ∪ *M*_{k} is contained in one of the quarters of *Q*_{k} obtained by cutting parallel to its sides (see Fig. 2 for an illustration of what was described above).

*x*∈

*Q*

_{0}and

*t*> 0. For the integral over

*L*

_{k}∪

*M*

_{k}, we use the fact that $H\Omega M(x,t)$, for

*x*∈

*L*

_{k}∪

*M*

_{k}and

*t*> 0, is bounded from below by $HQk(x,t)$. What one finds is

*d*(

*x*) = min{

*x*, 3 −

*x*},

*k*≥ 1,

*d*(

*x*

_{j}) interpreted appropriately,

*L*

_{k}∪

*M*

_{k}is contained in a quarter of the larger square

*Q*

_{k}, we have that

*e*

^{−1/(2t)}≤

*t*, for all

*t*≥ 0, we have proved that

## IV. AUXILIARY RESULTS

In this section, we prove a number of technical results that will be needed in the Proof of Theorem 1.1. Specifically, we prove that one can find a function whose *L*^{1} tail tends to zero slower than a given non-negative function *G* while satisfying some additional properties.

*Let*$G:(0,1)\u2192R$

*be a non-negative function with*$lim\tau \u21920+G(\tau )=0$

*. There exists a strictly decreasing non-negative and smooth function*

*h*∈

*L*

^{1}((0, 1))

*such that*

As a consequence of Proposition 4.1, we can prove the following result in the setting of sequences, which provides the final ingredient to complete the Proof of Theorem 1.1:

*Let*$G:(0,1)\u2192R$

*be a non-negative function with*$lim\tau \u21920+G(\tau )=0$

*and let*

*A*> 0

*. There exists a positive non-increasing sequence*${lk}k\u22651$

*with*

*∑*

_{k≥1}

*l*

_{k}=

*A*

*and a decreasing function*$M:(0,1)\u2192N$

*such that*

We first prove Proposition 4.1 and then show how to deduce Corollary 4.2 from it. To simplify the Proof of Proposition 4.1, it will be convenient to first show that we may assume that *G* is fairly well-behaved.

*Let* $G:(0,1)\u2192R$ *be a bounded non-negative function with* $lim\tau \u21920+G(\tau )=0$*. There exists a function* $G^\u2208C\u221e((0,1))$ *such that*

$lim\tau \u21920+G^(\tau )=0$

*,*$G^(\tau )>G(\tau )$

*for all**τ*∈ (0, 1)*,*$G^\u2032(\tau )>0$

*for all**τ*∈ (0, 1)*, and*$G^\u2032\u2032(\tau )<0$

*for all**τ*∈ (0, 1)*.*

*G*as in the lemma. Define, for

*τ*> 0,

*Ḡ*is non-decreasing, concave, and satisfies

*G*(

*τ*) ≤

*Ḡ*(

*τ*) for all

*τ*∈ (0, 1). This $lim\tau \u21920+\u1e20(\tau )=0$ follows if we can prove that for every

*b*> 0, there exists

*a*≥ 0 such that

*G*(

*s*) ≤

*as*+

*b*for all

*s*∈ [0, 1]. Since

*G*(

*s*) −

*b*is negative for

*s*small enough, the choice $a=sup0<s<1(G(s)\u2212b)+s$ works.

*φ*≥ 0,

*suppφ*⊆ [1, 2], and

*Ḡ*is non-decreasing,

*Ḡ*. Indeed, for 0 <

*τ*

_{1}<

*τ*

_{2}≤ 1,

*α*∈ (0, 1),

Clearly we can assume that *G* is bounded. By Lemma 4.3, we can without loss of generality assume that *G* is smooth, strictly increasing, and concave.

*h*(

*τ*) =

*G*′(

*τ*)

*U*(

*G*(

*τ*)) for some non-negative, integrable, and strictly decreasing

*U*∈

*C*

^{∞}((0, ‖

*G*‖

_{∞})) to be specified. By the assumptions on

*G*,

*U*, we have

*h*∈

*C*

^{∞}and, for 0 <

*τ*

_{1}<

*τ*

_{2}≤ 1,

*h*is strictly decreasing. Moreover,

*h*is integrable since

*U*so that $\u222b0sU(y)\u2009dy\u226bs$.

*G*is increasing and concave, we can bound

*U*≥ 0, the inequality (28) implies that

*U*. Using again (28) to bound the first term in the brackets, one finds

*y*∈ (0, ‖

*G*‖

_{∞}),

*G*(

*τ*) ≥

*τ*. By Proposition 4.1, there exists a non-negative strictly decreasing smooth function

*h*∈

*L*

^{1}((0, 1)) such that

*G*(

*τ*) ≥

*τ*implies that $lim\tau \u21920+h(\tau )=\u221e$.

*h*is strictly decreasing, we can consider its inverse, $h\u22121:(h(1),\u221e)\u2192R$, which is again strictly decreasing. Define $f:(0,\u221e)\u2192R$ by $f(y)=h\u22121(y)1y>h(1)+1y\u2264h(1)$. Then,

*h*is integrable and monotonic to conclude that $lim\tau \u21920+\tau h(\tau )=0$. Since the equation

*f*(

*y*) =

*τ*has a unique solution for each

*τ*∈ (0, 1), the inverse of $f\u22121:(0,1)\u2192R$ is well-defined, and by construction,

*f*

^{−1}(

*τ*) =

*h*(

*τ*).

Since *f* is decreasing and integrable, *∑*_{k≥1}*f*(*k*) < *∞*. Set *l*_{k} = *c*_{0}*f*(*k*) with *c*_{0} chosen so that *∑*_{k≥1}*l*_{k} = *A*. Define *M*(*τ*) to be the smallest integer such that *M*(*τ*) ≥ *f*^{−1}(*τ*) = *h*(*τ*). To complete the proof of the corollary, we need to relate the quantities in the statement to the corresponding quantities in Proposition 4.1.

*f*is monotonically decreasing, we can estimate

*x*=

*f*(

*y*), i.e.,

*y*=

*h*(

*x*), and an integration by parts,

*h*in Proposition 4.1 and

*G*(

*τ*) ≥

*τ*, we find

*M*(

*τ*) ≤

*h*(

*τ*) + 1,

## V. PROOF OF THEOREM 1.2

In the final section of this paper, we provide a proof that Theorem 1.1 implies Theorem 1.2.

Fix *R* as in the theorem. Without loss of generality, we may assume that *R* is bounded.

*C*<

*∞*,

*γ*≥ 0, and 0 ≤

*λ*

_{0}<

*∞*such that for all

*λ*≥

*λ*

_{0},

*μ*≥ 0,

*c*(

*t*) by discarding the negative volume term and use monotonicity to find

_{0}. This completes the Proof of Theorem 1.2.□

## ACKNOWLEDGMENTS

We would like to dedicate this paper to the memory of Jean Bourgain. The U.S. National Science Foundation (Grant No. DMS-1363432) (R.L.F.) and Knut and Alice Wallenberg Foundation (Grant No. KAW 2018.0281) (S.L.) are acknowledged. The authors also wish to thank Institut Mittag-Leffler, where part of this work was carried out.

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

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