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Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations

Priyank Oza, Jagmohan Tyagi

Abstract

We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, $s\in\big(\frac{1}{2},1\big).$ Due to inapplicability of comparison principle for the equations under consideration, one can not use the classical Perron's method for the existence of a solution. However, using the Lipschitz estimates established in theorem and vanishing viscosity method, we get the existence of solution. We further prove interior $C^{1,δ}$ regularity of the viscosity solutions using an improvement of the flatness technique when $s$ is close enough to $1.$

Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations

Abstract

We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, Due to inapplicability of comparison principle for the equations under consideration, one can not use the classical Perron's method for the existence of a solution. However, using the Lipschitz estimates established in theorem and vanishing viscosity method, we get the existence of solution. We further prove interior regularity of the viscosity solutions using an improvement of the flatness technique when is close enough to
Paper Structure (4 sections, 12 theorems, 191 equations)

This paper contains 4 sections, 12 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of main results
  4. Funding and/or Conflicts of interests/Competing interests

Key Result

Theorem 1.1

Let H3, H1 and H2 hold. Let $f\in L^\infty(B_1,\mathbb{R})$ with $\|f\|_{\infty,B_1}\leq \varepsilon$ for some $\varepsilon>0$ and $s\in ({1}/{2},1).$ Then for each $p\in\mathbb{R}^N,$ any bounded viscosity solution $u$ of with $\|u\|_{\infty,B_1}\leq 1$ is Lipschitz continuous with Lipschitz constant independent of $p.$ Moreover, the constant is uniformly bounded as $s\longrightarrow 1^-.$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Remark 3.5 C1beta
  • Corollary 1.4: Theorem 2.1 Bronzi
  • Corollary 1.5: Theorem 1.2 Topp
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 10 more