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

Abstract

We consider the Wulff-type energy functional,

$$\fancyscript{W}_\Omega(u) := \int_\Omega B( H( \nabla u (x) ) ) - F(u(x)) \, dx,$$

where B is positive, monotone and convex, and H is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate quasilinear equation in an anisotropic medium.

We prove that the gradient of the solution is bounded at any point by the potential F(u) and we deduce several rigidity and symmetry properties.

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Correspondence to Matteo Cozzi.

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Communicated by L.Caffarelli

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Cozzi, M., Farina, A. & Valdinoci, E. Gradient Bounds and Rigidity Results for Singular, Degenerate, Anisotropic Partial Differential Equations. Commun. Math. Phys. 331, 189–214 (2014). https://doi.org/10.1007/s00220-014-2107-9

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Keywords

  • Anisotropic Medium
  • Gradient Bound
  • Anisotropic Setting
  • Minimal Surface Equation
  • Anisotropic Surface Energy