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Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$

João Vítor da Silva, Julio Rossi, Ariel Salort

TL;DR

The paper analyzes dead-core phenomena for the $p$-Laplacian-type diffusion with strong absorption and develops sharp regularity estimates near the free boundary for fixed $p$ and $q$. It then studies the asymptotic regime $p \to \infty$, deriving a limiting fully nonlinear operator $\max\{-\Delta_{\infty} u, -|\nabla u|+u^{\ell}\}$ that governs the limit problem and obtaining explicit growth exponents $1/(1-\ell)$ and related non-degeneracy and density results. The work also proves convergence of free boundaries in the Hausdorff sense, establishes porosity and regularity properties of limit free boundaries, and provides average gradient bounds and $L^2$-type Hessian estimates to quantify behavior near the dead-core. Together these results advance understanding of dead-core formation, free boundary regularity, and the p to infinity transition with applications to reaction-diffusion and related models.

Abstract

We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$ -Δ_p u(x) + λ_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad Ω\subset \mathbb{R}^N, $$ where $λ_0>0$ is a bounded function, $Ω$ is a bounded domain and $0\leq q<p-1$. When $p$ is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for $p-$dead core solutions. Afterwards, assuming that $\ell \:=\lim_{p \to \infty} q(p)/p \in [0, 1)$ exists, we establish existence for limit solutions as $p\to \infty$, as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp $C^γ$ regularity estimates for limit solutions along free boundary points, that is, points on $ \partial \{u>0\} \cap Ω$ where the sharp regularity exponent is given explicitly by $γ= \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.

Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$

TL;DR

The paper analyzes dead-core phenomena for the -Laplacian-type diffusion with strong absorption and develops sharp regularity estimates near the free boundary for fixed and . It then studies the asymptotic regime , deriving a limiting fully nonlinear operator that governs the limit problem and obtaining explicit growth exponents and related non-degeneracy and density results. The work also proves convergence of free boundaries in the Hausdorff sense, establishes porosity and regularity properties of limit free boundaries, and provides average gradient bounds and -type Hessian estimates to quantify behavior near the dead-core. Together these results advance understanding of dead-core formation, free boundary regularity, and the p to infinity transition with applications to reaction-diffusion and related models.

Abstract

We study regularity issues and the limiting behavior as of nonnegative solutions for elliptic equations of Laplacian type () with a strong absorption: where is a bounded function, is a bounded domain and . When is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for dead core solutions. Afterwards, assuming that exists, we establish existence for limit solutions as , as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp regularity estimates for limit solutions along free boundary points, that is, points on where the sharp regularity exponent is given explicitly by . Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.
Paper Structure (9 sections, 29 theorems, 177 equations, 1 figure)

This paper contains 9 sections, 29 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularity properties for fixed $(p,q)$
  4. Non-degeneracy of solutions
  5. Growth rate near of free boundary
  6. Applications
  7. The limit problem
  8. Further properties for limit solutions
  9. Final comments

Key Result

Theorem 1.1

Let $u$ be a nonnegative, bounded weak solution to Eqp-Lapla, $\Omega^{\prime} \Subset \Omega$ and let $x_0 \in \overline{\{u >0\}} \cap \Omega^{\prime}$. Then there exists a universal constant $\mathfrak{C}_0>0$ such that for all $0<r<\min\{1, \mathrm{dist}(\Omega^{\prime}, \partial \Omega)\}$ ther

Figures (1)

  • Figure 1: The dead-core set $\Omega_0$ illustrating the isothermal and irreversible catalytical reaction process from \ref{['ExChemCat']}.

Theorems & Definitions (67)

  • Theorem 1.1: Strong non-degeneracy
  • Theorem 1.2: Improved regularity along the free boundary
  • Theorem 1.3: Limiting equation
  • Theorem 1.4: Sharp growth for limit solutions
  • Theorem 1.5: Strong non-degeneracy for limit solutions
  • Theorem 1.6: An average gradient estimate for limit solutions
  • Definition 2.2: Weak solution
  • Definition 2.3: Viscosity solution
  • Definition 2.4: Viscosity solution for the limit equation
  • Theorem 2.5: Serrin's Harnack inequality Ser64
  • ...and 57 more