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Optimal $C^{1,α}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

Junior da Silva Bessa, Jehan Oh

Abstract

In this paper we establish optimal $C^{1,α}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary Hölder estimates for perturbed models with oblique boundary data in ``almost $C^{1}$-flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal Hölder exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.

Optimal $C^{1,α}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

Abstract

In this paper we establish optimal regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary Hölder estimates for perturbed models with oblique boundary data in ``almost -flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal Hölder exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.

Paper Structure

This paper contains 8 sections, 9 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Some implications
  4. Preliminaries and auxiliary results
  5. Optimal C1,nu Regularity up to the Boundary: Proof of Theorem 1.2
  6. Universal improvement of flatness
  7. Proof of the Main Theorem
  8. Improved regularity at vanishing source points: Proof of Theorem \ref{['Thm2']}

Key Result

Theorem 1.2

Let $\Omega\subset \mathbb{R}^{n}$ be a $C^{1}$-bounded domain with $0\in \partial \Omega_{1}$. Suppose $u$ is a viscosity solution to satisfying the structural conditions (A1)--(A3). Then, $u\in C^{1,\nu}(\overline{\Omega_{\frac{1}{2}}})$ with Moreover, the following estimate holds where $\mathrm{C}>0$ depends only on $n$, $\lambda$, $p$, $q$, $\alpha_{0}$, $\delta_{0}$, $[\beta]_{C^{0,\alpha}

Theorems & Definitions (19)

  • Remark 1.1
  • Theorem 1.2: Optimal $C^{1,\alpha}$ Regularity
  • Remark 1.3: One-phase degeneracy
  • Corollary 1.4
  • Theorem 1.5: Improved regularity along vanishing source points
  • Definition 2.1: Viscosity solution
  • Theorem 2.2
  • Theorem 2.3: Hölder regularity
  • proof
  • Lemma 2.4: Cutting Lemma
  • ...and 9 more