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Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients

About this Title

Lyudmila Korobenko, Cristian Rios, Eric Sawyer and Ruipeng Shen

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 269, Number 1311
ISBNs: 978-1-4704-4401-3 (print); 978-1-4704-6460-8 (online)
DOI: https://doi.org/10.1090/memo/1311
Published electronically: March 2, 2021
Keywords: Regularity, quasilinear equations, infinite degeneracy, rough coefficients, non-doubling control balls, subrepresentation inequality, Orlicz-Sobolev inequality

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Table of Contents

Chapters

• Preface

1. Overview

• 1. Introduction
• 2. DeGiorgi Iteration, Local Boundedness, Maximum Principle and Continuity
• 3. Organization of the Proofs

2. Abstract theory

• 4. Local Boundedness
• 5. Maximum Principle
• 6. Continuity

3. Geometric theory

• 7. Infinitely Degenerate Geometries
• 8. Orlicz Norm Sobolev Inequalities
• 9. Geometric Theorems

A. Appendix

• A. A Quasilinear Hypoellipticity Theorem
• B. A Monge-Ampère Example
• C. Absence of Radial Solutions

Abstract

We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form inhomogeneous equations, and also continuity of weak solutions to homogeneous equations. For example, we consider the family $\left \{ f_{\sigma }\right \} _{\sigma >0}$ with \begin{equation*} f_{\sigma }\left ( x\right ) =e^{-\left ( \frac {1}{\left \vert x\right \vert }\right ) ^{\sigma }},\ \ \ \ \ -\infty <x<\infty , \end{equation*} of infinitely degenerate functions at the origin, and show that all weak solutions to the associated infinitely degenerate quasilinear equations of the form \begin{equation*} \mathrm {div}A\left ( x,u\right ) \mathrm {grad}u=\phi \left ( x\right ) ,\ \ \ A\left ( x,z\right ) \sim \left [ \begin {array}{cc} I_{n-1} & 0 \\ 0 & f\left ( x_{1}\right ) ^{2}\end{array}\right ] , \end{equation*} with rough data $A$ and $\phi$, are locally bounded for admissible $\phi$ provided $0<\sigma <1$. We also show that these conditions are necessary for local boundedness in dimension $n\geq 3$, thus paralleling the known theory for the smooth Kusuoka-Strook operators $\frac {\partial ^{2}}{\partial x_{1}^{2}}+\frac {\partial ^{2}}{\partial x_{2}^{2}}+f_{\sigma }\left ( x\right ) ^{2}\frac {\partial ^{2}}{\partial x_{3}^{2}}$. We also show that subsolutions satisfy a maximum principle under the same restriction on the degeneracy. Finally, continuity of solutions is derived in the homogeneous case $\phi \equiv 0$ under a more stringent assumption on the degeneracy, namely that $f\geq f_{3,\sigma }$ for $0<\sigma <1$ where\begin{equation*} f_{3,\sigma }\left ( x\right ) =\left \vert x\right \vert ^{\left ( \ln \ln \ln \frac {1}{\left \vert x\right \vert }\right ) ^{\sigma }},\ \ \ \ \ -\infty <x<\infty . \end{equation*}

As an application we obtain weak hypoellipticity (i.e. smoothness of all weak solutions) of certain infinitely degenerate quasilinear equations in the plane \begin{equation*} \frac {\partial u}{\partial x^{2}}+f\left ( x,u\left ( x,y\right ) \right ) ^{2}\frac {\partial u}{\partial y^{2}}=0, \end{equation*} with smooth data $f\left ( x,z\right ) \gtrsim f_{3,\sigma }\left ( x\right )$ where $f\left ( x,z\right )$ has a sufficiently mild nonlinearity and degeneracy.

In order to prove these theorems, we first establish abstract results in which certain Poincaré and Orlicz Sobolev inequalities are assumed to hold. We then develop subrepresentation inequalities for control geometries in order to obtain the needed Poincaré and Orlicz Sobolev inequalities.