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A two-phase quenching-type problem for the p-Laplacian

Julio C. Correa, Disson dos Prazeres

TL;DR

The paper analyzes minimizers of a non-differentiable two-phase $p$-Laplacian functional with $p>2$, focusing on the geometry and regularity of the free boundary between the positive and negative phases. It develops variational solutions, establishes non-degeneracy and stability as $p\to 2$, and proves sharp optimal growth near free-boundary points; it also shows the free boundary has bounded $(n-1)$-dimensional Hausdorff measure. In 2D, the results hold without the near-2 restriction on $p$ thanks to enhanced $p$-harmonic regularity and a Liouville-type argument, yielding the same sharp growth and perimeter conclusions. Collectively, the work extends Weiss-type blow-up classifications from the linear case to a nonlinear $p$-Laplacian setting and provides strong structure theorems for the two-phase free boundary problem with Hölder potentials.

Abstract

We study minimizers of non-differentiable functionals of the Alp-Phillips type with two-phases for the $p$-Laplacian , focusing on the geometric and analytical properties of free boundaries. The main result establishes finite $(n-1)$-dimensional Hausdorff measure estimates, achieved through optimal gradient decay estimates, a $BV$-inequality and the known classifications of blow-up profiles of the linear case.

A two-phase quenching-type problem for the p-Laplacian

TL;DR

The paper analyzes minimizers of a non-differentiable two-phase -Laplacian functional with , focusing on the geometry and regularity of the free boundary between the positive and negative phases. It develops variational solutions, establishes non-degeneracy and stability as , and proves sharp optimal growth near free-boundary points; it also shows the free boundary has bounded -dimensional Hausdorff measure. In 2D, the results hold without the near-2 restriction on thanks to enhanced -harmonic regularity and a Liouville-type argument, yielding the same sharp growth and perimeter conclusions. Collectively, the work extends Weiss-type blow-up classifications from the linear case to a nonlinear -Laplacian setting and provides strong structure theorems for the two-phase free boundary problem with Hölder potentials.

Abstract

We study minimizers of non-differentiable functionals of the Alp-Phillips type with two-phases for the -Laplacian , focusing on the geometric and analytical properties of free boundaries. The main result establishes finite -dimensional Hausdorff measure estimates, achieved through optimal gradient decay estimates, a -inequality and the known classifications of blow-up profiles of the linear case.

Paper Structure

This paper contains 9 sections, 13 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The problem
  3. Known results
  4. Main Results
  5. Existence of Variational Solutions
  6. Non-Degeneracy (Stability)
  7. Optimal growth and its consequences
  8. $\mathcal{H}^{n-1}$-estimates on the free boundary
  9. The two-dimensional case

Key Result

Theorem 1

Given $0<\gamma<p/2$ and $u\in \mathcal{P}_1(M,p,\gamma,y)$. There exist positive constants $C$, $\delta$, and $r_o$, depending on $\lambda_i$, $M$, and the dimension, such that: if $\left\lvert p-2\right\rvert<\delta$ then for all $r<r_o$ and $y\in B_{1/2}\cap\Gamma(u)$.

Figures (1)

  • Figure 1: $\Gamma^+(u):=\partial\{u>0\}\cap B_R(x_0)$, $\Gamma^-(u):\partial\{u<0\}\cap B_R(x_o)$, $x_o$ branching two-phase point, $x_1$ non-branching two-phase point, $x_2$ and $x_3$ one-phase points. The red portion represents a typical neighbourhood of a branching two-phase point.

Theorems & Definitions (34)

  • Theorem 1: Optimal Growth
  • Theorem 2: $\mathcal{H}^{n-1}$- Estimates on the Free Boundary
  • Theorem 3
  • Definition 1
  • Remark 1
  • Proposition 1
  • proof
  • Definition 2
  • Remark 2
  • Proposition 2
  • ...and 24 more