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Fully Nonlinear Singularly perturbed models with non-homogeneous degeneracy

João V. Silva, Elzon C. Júnior, Gleydson C. Ricarte

Abstract

In our work we study non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In such a scenario, we establish the existence of solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Moreover, for a restricted class of non-linearities, we prove the finiteness of the (N-1)-dimensional Hausdorff measure of level sets. We also address a complete analysis concerning the asymptotic limit as the singular parameter, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory.

Fully Nonlinear Singularly perturbed models with non-homogeneous degeneracy

Abstract

In our work we study non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In such a scenario, we establish the existence of solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Moreover, for a restricted class of non-linearities, we prove the finiteness of the (N-1)-dimensional Hausdorff measure of level sets. We also address a complete analysis concerning the asymptotic limit as the singular parameter, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory.

Paper Structure

This paper contains 21 sections, 34 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Main assumptions
  3. Statement of main results
  4. Motivations and State-of-Art
  5. Modern developments in singular perturbation theory
  6. Recent progresses on degenerate equations in non-divergence form
  7. Background results
  8. Lipschitz regularity estimates
  9. Geometric non-degeneracy
  10. Building Barriers
  11. Linear Growth
  12. Some important implications from Theorems \ref{['thmreg']} and \ref{['cresc']}
  13. A Harnack type inequality
  14. Porosity of the level surfaces
  15. Hausdorff measure estimates
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and let $0 \le g \in C(\partial \Omega)$ be a boundary datum. Then, for each fixed $\varepsilon>0$ there exists a non-negative viscosity solution $u^{\varepsilon} \in C(\overline{\Omega})$ to Equation Pe.

Theorems & Definitions (67)

  • Definition \oldthetheorem: Viscosity solution
  • Definition \oldthetheorem: Perron type solution
  • Theorem \oldthetheorem: Existence of Perron solutions
  • Theorem \oldthetheorem: Optimal Lipschitz estimate
  • Theorem \oldthetheorem: Linear growth
  • Theorem \oldthetheorem: Strong Non-degeneracy
  • Theorem \oldthetheorem: The limiting PDE
  • Theorem \oldthetheorem: Asymptotic behavior close free boundary
  • Theorem \oldthetheorem: Hausdorff estimates
  • Theorem \oldthetheorem: $C_{\text{loc}}^{1, \alpha}-$estimates
  • ...and 57 more