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Lipschitz regularity of solutions to two-phase $p$-Laplacian free boundary problems with right hand side

Fausto Ferrari, Claudia Lederman

Abstract

We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the $p$-Laplacian with non-zero right hand side, where $p\in (1,\infty)$. This is the optimal regularity for the problem. We also obtain the local Hölder continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.

Lipschitz regularity of solutions to two-phase $p$-Laplacian free boundary problems with right hand side

Abstract

We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the -Laplacian with non-zero right hand side, where . This is the optimal regularity for the problem. We also obtain the local Hölder continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.
Paper Structure (6 sections, 13 theorems, 186 equations)

This paper contains 6 sections, 13 theorems, 186 equations.

Table of Contents

  1. Introduction and main results
  2. Assumptions
  3. Basic definitions, notation and preliminaries
  4. Regularity of flat free boundaries
  5. Hölder continuity
  6. Lipschitz continuity

Key Result

Theorem 1.1

Let $u$ be a viscosity solution to fbtrue in $B_1$. Assume that assump-f holds in $B_1$ and $G$ satisfies assumptions (P1), (P2) and (P3) in $B_1$. If $f\not\equiv 0$ and $p\neq 2$ assume that also (P4) holds, then where $C=C(n,p,\|f\|_{L^\infty(B_1)}, G)$ is a positive constant.

Theorems & Definitions (31)

  • Theorem 1.1: Optimal regularity
  • Theorem 1.2: Local Hölder continuity
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Definition 2.7: Definition 7.1 in FL3
  • Theorem 2.8
  • Definition 2.10
  • Theorem 3.1: Flatness implies $C^{1,\gamma}$
  • ...and 21 more