Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal regularity estimates for solutions to fully nonlinear elliptic equations with oblique boundary data

Junior da S. Bessa, João Vitor da Silva, Gleydson C. Ricarte

Abstract

In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by $$ \left\{ \begin{array}{rcl} F(D^2u,x) &=& f(x) \quad \mbox{in} \,\, Ω\\ β(x) \cdot Du(x) + γ(x) \, u(x)&=& g(x) \quad \mbox{on} \,\, \partial Ω. \end{array} \right. $$ Such regularity estimates are achieved by exploring the integrability properties of $f$ based on different scenarios, making a $\text{VMO}$ assumption on the coefficients of $F$, and by considering suitable smoothness properties on the boundary data $β, γ$ and $g$. Particularly, we derive sharp estimates for borderline cases where $f \in L^n(Ω)$ and $f\in p-\textrm{BMO}(Ω)$. Additionally, for source terms in $L^p(Ω)$, for $p \in (n, \infty)$, we obtain sharp gradient estimates. Finally, we also address Schauder-type estimates for convex/concave operators and suitable Hölder data.

Universal regularity estimates for solutions to fully nonlinear elliptic equations with oblique boundary data

Abstract

In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by Such regularity estimates are achieved by exploring the integrability properties of based on different scenarios, making a assumption on the coefficients of , and by considering suitable smoothness properties on the boundary data and . Particularly, we derive sharp estimates for borderline cases where and . Additionally, for source terms in , for , we obtain sharp gradient estimates. Finally, we also address Schauder-type estimates for convex/concave operators and suitable Hölder data.
Paper Structure (7 sections, 17 theorems, 267 equations, 2 tables)

This paper contains 7 sections, 17 theorems, 267 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Assumptions and auxiliary results
  3. Optimal Hölder estimates
  4. Log-Lipschitz regularity estimates
  5. Optimal gradient regularity
  6. $C^{1, Log-Lip}$ regularity estimates
  7. Schauder-type estimates

Key Result

Lemma 2.3

For $k \in \mathbb{N}$ let $\Omega_k \subset \Omega_{k+1}$ be an increasing sequence of domains and $\Omega \mathrel{\mathop:}= \bigcup_{k=1}^{\infty} \Omega_k$. Let $F, F_k$ be $(\lambda, \Lambda)-$elliptic operators. Assume $f \in L^{p}(\Omega)$, $f_k \in L^p(\Omega_k)$ and that $u_k \in C^0(\Omeg Suppose that $u_k \to u_{\infty}$ locally uniformly in $\Omega$ and that for $\mathrm{B}_r(x_0) \su

Theorems & Definitions (44)

  • Remark 2.1
  • Definition 2.2: $C^2$-viscosity solution
  • Lemma 2.3: Stability Lemma
  • Definition 2.4
  • Lemma 2.5: A.B.P. Maximum Principle
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10: Morrey spaces
  • ...and 34 more