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Regularity of solutions to degenerate and singular free boundary problems with volume constraint

T. M. Nascimento, X. H. Nguyen, P. R. Stinga

Abstract

We prove existence and regularity of solutions to degenerate and singular elliptic free boundary problems, where the volume of the positivity set of the solution is prescribed.

Regularity of solutions to degenerate and singular free boundary problems with volume constraint

Abstract

We prove existence and regularity of solutions to degenerate and singular elliptic free boundary problems, where the volume of the positivity set of the solution is prescribed.

Paper Structure

This paper contains 4 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The penalized problem with parameter $\varepsilon>0$
  4. Behavior for small $\varepsilon$

Key Result

Theorem 1.1

Let $\omega$ be an $A_2$ weight. Assume that either $\omega$ satisfies the isoperimetric inequality eq:isoperimetricassumption, or that it is bounded below away from zero in $B_1$ as in eq:lowerboundassumption. Then there is a minimizer $u\in K_0$ to problem weighted prob such that $0\leq u\leq\|g\| where $C>0$ depends only on $K$, $[\omega]_{A_2}$ and $n$. Furthermore, $u$ is a weak solution to $

Theorems & Definitions (17)

  • Theorem 1.1: Existence and regularity of solutions
  • Lemma 2.1: Properties of $A_2$ weights
  • Lemma 2.2: Weighted Poincaré inequality
  • Theorem 2.3: Harnack inequality and Hölder regularity
  • Lemma 3.1: Existence of minimizers for \ref{['penalized problem']}
  • proof
  • Lemma 3.2: Boundedness of minimizers
  • proof
  • Definition 3.3
  • Theorem 3.4
  • ...and 7 more