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Geometric properties of free boundaries in degenerate quenching problems

Damião J. Araújo, Rafayel Teymurazyan, José Miguel Urbano

TL;DR

This work analyzes the free boundary geometry in degenerate quenching problems driven by the $p$-Laplace energy with a non-differentiable Alt–Phillips type potential $F(x,s)=[s]_+^{\gamma}$, where $\gamma\in(0,1)$. The authors develop sharp gradient-decay estimates via an intrinsic Harnack inequality and establish a universal flatness regime to obtain refined regularity near the interface; these tools culminate in a finite $(n-1)$-dimensional Hausdorff measure for $\partial\{u>0\}$ and porosity-based measure-zero results for the free boundary. A nondegeneracy/density analysis, together with a Besicovitch-covering argument, yields a local perimeter bound and, up to a negligible set, a decomposition of the free boundary into a countable union of $C^1$ hypersurfaces. The approach extends the $p=2$ obstacle/cavity theory to the nonlinear, degenerate regime $p>2$, and provides alternative proofs of classical results (e.g., Phillips) with potential applicability to more general free boundary problems.

Abstract

We investigate minimizers of non-differentiable functionals epitomizing the degenerate quenching problem. The main result establishes the finiteness of the $(n-1)-$dimensional Hausdorff measure of the free boundary. Our approach hinges on optimal gradient decay estimates, derived via an intrinsic Harnack-type inequality, combined with a fine analysis in a flatness regime, where minimizers exhibit enhanced regularity. While the results are formulated for the degenerate setting, they also provide insights applicable to the singular case. Additionally, the findings offer an alternative proof of classical results by Phillips, further demonstrating their robustness and potential for broader applicability in the analysis of free boundary problems.

Geometric properties of free boundaries in degenerate quenching problems

TL;DR

This work analyzes the free boundary geometry in degenerate quenching problems driven by the -Laplace energy with a non-differentiable Alt–Phillips type potential , where . The authors develop sharp gradient-decay estimates via an intrinsic Harnack inequality and establish a universal flatness regime to obtain refined regularity near the interface; these tools culminate in a finite -dimensional Hausdorff measure for and porosity-based measure-zero results for the free boundary. A nondegeneracy/density analysis, together with a Besicovitch-covering argument, yields a local perimeter bound and, up to a negligible set, a decomposition of the free boundary into a countable union of hypersurfaces. The approach extends the obstacle/cavity theory to the nonlinear, degenerate regime , and provides alternative proofs of classical results (e.g., Phillips) with potential applicability to more general free boundary problems.

Abstract

We investigate minimizers of non-differentiable functionals epitomizing the degenerate quenching problem. The main result establishes the finiteness of the dimensional Hausdorff measure of the free boundary. Our approach hinges on optimal gradient decay estimates, derived via an intrinsic Harnack-type inequality, combined with a fine analysis in a flatness regime, where minimizers exhibit enhanced regularity. While the results are formulated for the degenerate setting, they also provide insights applicable to the singular case. Additionally, the findings offer an alternative proof of classical results by Phillips, further demonstrating their robustness and potential for broader applicability in the analysis of free boundary problems.
Paper Structure (6 sections, 17 theorems, 156 equations, 1 figure)

This paper contains 6 sections, 17 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setting and preliminary results
  3. A pointwise gradient estimate
  4. Non-degeneracy and porosity
  5. Universal flatness regime
  6. Hausdorff measure estimates

Key Result

Lemma 2.1

Let $B\subset\Omega$ be a ball, and let $u$ be a minimizer of 1.1--minimizationset. Consider the $p-$harmonic replacement of $u$, i.e., the $p-$harmonic function $v$ in $B$ that agrees with $u$ on $\partial B$. Then $u\le v$ in $B$.

Figures (1)

  • Figure 1: Behaviour of minimizers in the flatness regime.

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Theorem 3.1
  • ...and 28 more