Overdetermined anisotropic elliptic problems

Abstract

A symmetry result is established for solutions to overdetermined anisotropic elliptic problems in variational form, which extends Serrin's theorem dealing with the isotropic radial case. The involved anisotropy arises from replacing the Euclidean norm of the gradient with an arbitrary norm in the associated variational integrals. The resulting symmetry of the solutions is that of the so-called Wulff shape.

Gradient Bounds and Rigidity Results for Singular, Degenerate, Anisotropic Partial Differential Equations

“…[NP99,BNP01a,BNP01b,EO04,OBGXY05] and references therein. See also [FM91,C04] for anisotropic problems related to the Willmore functional and [CS09,WX11] for elliptic anisotropic systems inspired by fluidodynamics. We defer the interested reader to Appendix C for some deeper physical insights.…”

A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

“…By the way, we hope the method introduced in this paper can be beneficial to the wellposedness problems for the evolutionary ( )-Laplacian equations, overdetermined anisotropic elliptic equations, and the infiltration equations. As we know, there are many papers devoted to these equations; one can refer to [11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein.…”

An overdetermined problem for the anisotropic capacity

“…Notice that, in both problems (4) and (2)- (3), the radial symmetry of the solution is compelled by the isotropy of the Euclidean norm and of the Laplacian. Considering in particular problem (2), we see that the Laplace operator reflects the linearity of the electrical conduction law, which is in turn determined by the isotropy of the dielectric and dictates the use of the Euclidean norm in measuring the electric field in condition (3).…”

“…In [6] the authors improve the results of [22], weakening the regularity assumptions on the set . In [4], the authors consider the anisotropic version of the classical Serrin's problem (4)…”

“…Serrin proved that a solution to problem (4) exists if and only if is an Euclidean ball (and hence u is radially symmetric). Notice that, in both problems (4) and (2)- (3), the radial symmetry of the solution is compelled by the isotropy of the Euclidean norm and of the Laplacian.…”