An overdetermined problem for the anisotropic capacity

Abstract

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in ${\mathbb{R}}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).

References in corpus (2)

• Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs
• Anisotropic elliptic equations with general growth in the gradient and Hardy-type potentials

Cited by in corpus (1)

• Radial symmetry for p-harmonic functions in exterior and punctured domains