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Regularity estimates for fully nonlinear dead-core problems with a Hamiltonian Term

Rafael R. Costa, Ginaldo S. Sá

TL;DR

This work analyzes a dead-core phenomenon for a fully nonlinear elliptic equation with a Hamiltonian term, $| abla u|^p F(D^2u) + \mathfrak{a}(x)|\nabla u|^q = f(x,u)$, in a smooth bounded domain, where plateau regions and a free boundary arise. It develops a barrier-based and geometric flatness iteration approach to obtain sharp pointwise $C^{1,\beta}$ regularity along the free boundary, with $\beta = \min\{\frac{p+2-q}{p+1-q}, \frac{p+2}{p+1-\mu}\}$, under $0<\mu<p+1$ and $0<q<p+1$, and proves a corresponding non-degeneracy result that identifies optimal growth rates on the free boundary. The paper also establishes a comparison principle, existence/uniqueness via Perron’s method, local Lipschitz estimates, and consequences such as positive density of the positivity set and a finite Hausdorff measure bound for the free boundary. Together, these results advance the understanding of regularity and geometric properties of dead-core solutions to degenerate/singular fully nonlinear problems with Hamiltonian terms, with potential applications to diffusion-reaction models and phase-transition phenomena.

Abstract

In this paper, we present a problem involving fully nonlinear elliptic operators with Hamiltonian, which can present a singularity or degenerate as the gradient approaches the origin. The model studied here, allows the appearance of plateau zones, i.e. unknown regions of the domain in which the non-negative solutions vanishes. We show an improvement in regularity along the free boundary of the problem, and with some hypotheses on the exponents of the equation we proved the optimality of the growth rate with the help of the non-degeneracy also obtained here. In addition, some more applications of the growth results on the free boundary are obtained, such as: the positivity of solutions and also information on the Hausdorff measure of the boundary of the coincidence set.

Regularity estimates for fully nonlinear dead-core problems with a Hamiltonian Term

TL;DR

This work analyzes a dead-core phenomenon for a fully nonlinear elliptic equation with a Hamiltonian term, , in a smooth bounded domain, where plateau regions and a free boundary arise. It develops a barrier-based and geometric flatness iteration approach to obtain sharp pointwise regularity along the free boundary, with , under and , and proves a corresponding non-degeneracy result that identifies optimal growth rates on the free boundary. The paper also establishes a comparison principle, existence/uniqueness via Perron’s method, local Lipschitz estimates, and consequences such as positive density of the positivity set and a finite Hausdorff measure bound for the free boundary. Together, these results advance the understanding of regularity and geometric properties of dead-core solutions to degenerate/singular fully nonlinear problems with Hamiltonian terms, with potential applications to diffusion-reaction models and phase-transition phenomena.

Abstract

In this paper, we present a problem involving fully nonlinear elliptic operators with Hamiltonian, which can present a singularity or degenerate as the gradient approaches the origin. The model studied here, allows the appearance of plateau zones, i.e. unknown regions of the domain in which the non-negative solutions vanishes. We show an improvement in regularity along the free boundary of the problem, and with some hypotheses on the exponents of the equation we proved the optimality of the growth rate with the help of the non-degeneracy also obtained here. In addition, some more applications of the growth results on the free boundary are obtained, such as: the positivity of solutions and also information on the Hausdorff measure of the boundary of the coincidence set.
Paper Structure (7 sections, 8 theorems, 145 equations)

This paper contains 7 sections, 8 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Main results and some consequences
  3. Hypothesis and definitions
  4. Existence/Uniqueness of viscosity solutions
  5. Local Lipschitz estimates
  6. Improved regularity estimate: Proof of Theorem \ref{['improvement_regularity']}
  7. Non-degeneracy estimates: Proof of Theorem \ref{['ND_estimates']}

Key Result

Theorem 1.1

Let $u$ be a non-negative and bounded viscosity solution to equation and consider $x_0\in\partial\{u>0\}\cap\Omega'$, where $\Omega'\Subset\Omega$. Then, for $0<r<\min\left\{1,\frac{\operatorname{dist}(\Omega',\partial\Omega)}{2}\right\}$ and any $x\in B_r(x_0)\cap\{u>0\}$, there holds where and $\mathrm{C}_0$ is a constant depending only on $n,\lambda,\Lambda,p,q,\mu,\|\mathfrak{a}\|_{L^{\infty

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2: Non-degeneracy estimate
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1: Comparison Principle
  • proof
  • Theorem 3.2: Existence and uniquiness
  • ...and 9 more