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Interior $W^{2,δ}$ type estimates for degenerate fully nonlinear elliptic equations with $L^n$ data

Sun-Sig Byun, Hongsoo Kim, Jehan Oh

Abstract

We establish interior $W^{2,δ}$ type estimates for a class of degenerate fully nonlinear elliptic equations with $L^n$ data. The main idea of our approach is to slide $C^{1,α}$ cones, instead of paraboloids, vertically to touch the solution, and estimate the contact set in terms of the measure of the vertex set. This shows that the solution has tangent $C^{1,α}$ cones almost everywhere, which leads to the desired Hessian estimates. Accordingly, we are able to develop a kind of counterpart to the estimates for divergent structure quasilinear elliptic problems.

Interior $W^{2,δ}$ type estimates for degenerate fully nonlinear elliptic equations with $L^n$ data

Abstract

We establish interior type estimates for a class of degenerate fully nonlinear elliptic equations with data. The main idea of our approach is to slide cones, instead of paraboloids, vertically to touch the solution, and estimate the contact set in terms of the measure of the vertex set. This shows that the solution has tangent cones almost everywhere, which leads to the desired Hessian estimates. Accordingly, we are able to develop a kind of counterpart to the estimates for divergent structure quasilinear elliptic problems.

Paper Structure

This paper contains 4 sections, 10 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The relation between a tangent $C^{1,\alpha}$ cone and $D(|Du|^\gamma Du$)
  4. Proof of Theorem \ref{['tangent']}

Key Result

Theorem 1.1

Let $u \in C(\overline{B}_{1})$ satisfy in the viscosity sense, with $0 < \lambda \leq \Lambda < \infty$, $\gamma \geq 0$ and $f \in C \cap L^{n}(B_{1})$. Then $|Du|^\gamma Du \in W^{1,\delta}(B_{1/2})$ with the estimate where $\delta>0$ and $C>0$ depend only on $n, \lambda, \Lambda$ and $\gamma$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • ...and 7 more