We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.
REFERENCES
1.
L. M.
Sibner
and R. J.
Sibner
, “A nonlinear Hodge-de Rham theorem
,” Acta Math.
125
, 57
–73
(1970
).2.
L. M.
Sibner
and R. J.
Sibner
, “Nonlinear Hodge theory: Applications
,” Adv. Math.
31
, 1
–15
(1979
).3.
L. M.
Sibner
and R. J.
Sibner
, “A maximum principle for compressible flow on a surface
,” Proc. Am. Math. Soc.
71
, 103
–108
(1978
).4.
L. M.
Sibner
and R. J.
Sibner
, “A subelliptic estimate for a class of invariantly defined elliptic systems
,” Pac. J. Math
94
, 417
–421
(1981
).5.
K.
Uhlenbeck
, “Regularity for a class of nonlinear elliptic systems
,” Acta Math.
138
, 219
–240
(1977
).6.
Y. Yang, “Classical solutions in the Born–Infeld theory,” preprint.
7.
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (Wiley, New York, 1958).
8.
T. H.
Otway
, “Yang–Mills fields with nonquadratic energy
,” J. Geom. Phys.
19
, 379
–398
(1996
).9.
T. H.
Otway
, “Properties of nonlinear Hodge fields
,” J. Geom. Phys.
27
, 65
–78
(1998
).10.
11.
R. Schoen, “Analytic aspects of the harmonic map problem,” in Seminar in Nonlinear Partial Differential Equations, edited by S. S. Chern (Springer-Verlag, New York, 1984).
12.
R.
Hardt
and F-H.
Lin
, “Mappings minimizing the norm of the gradient
,” Commun. Pure Appl. Math.
40
, 555
–588
(1987
).13.
L. M.
Sibner
, “An existence theorem for a nonregular variational problem
,” Manuscr. Math.
43
, 45
–72
(1983
).14.
J. Jost, Nonlinear Methods in Riemannian and Kählerian Geometry (Birkhäuser, Basel, 1988).
15.
S.
Takakuwa
, “On removable singularities of stationary harmonic maps
,” J. Fac. Sci., Univ. Tokyo, Sect. 1
32
, 373
–395
(1985
);S. Hildebrandt, in Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, edited by S. S. Chern and Wu Wen-tsün (Science Press, Beijing, and Gordon and Breach, New York, 1982).
16.
J.
Serrin
, “Local behavior of solutions of quasilinear equations
,” Acta Math.
111
, 247
–302
(1964
).17.
J.
Serrin
, “Removable singularities of solutions of elliptic equations
,” Arch. Ration. Mech. Anal.
17
, 67
–78
(1964
).18.
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (Princeton University Press, Princeton, 1983).
19.
C. B. Morrey, Multiple Integrals in the Calculus of Variations (Springer-Verlag, Berlin, 1966).
20.
G.
Liao
, “A regularity theorem for harmonic maps with small energy
,” J. Diff. Geom.
22
, 233
–241
(1985
).21.
B.
Gidas
and J.
Spruck
, “Global and local behavior of positive solutions of nonlinear elliptic equations
,” Commun. Pure Appl. Math.
34
, 525
–598
(1981
);T. H.
Otway
and L. M.
Sibner
, “Point singularities of coupled gauge fields with low energy
,” Commun. Math. Phys.
111
, 275
–279
(1987
).22.
D. G. B. Edelen, Applied Exterior Calculus (Wiley, New York, 1985).
23.
C. Carathéodory, Gesammelte Mathematische Schriften, Bd. II, S. 131-177 (C. H. Beck’sche Verlagsbuchhandlung, Münich, 1955).
24.
A.
Lichnerowicz
, “Courbure et nombres de Betti d’une varieté riemannienne compacte
,” C. R. Acad. Sci. Paris
226
, 1678
–1680
(1948
).25.
D. Yang, “ pinching and compactness theorems for compact Riemannian manifolds,” preprint.
26.
D.
Yang
, “Convergence of Riemannian manifolds with integral bounds on curvature I
,” Ann. Sci. École Norm. Superieure
25
, 77
–105
(1992
).
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.