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Fully nonlinear dead-core systems

Damião J. Araújo, Rafayel Teymurazyan

Abstract

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system and obtain higher sharp regularity across the free boundary. Additionally, we prove geometric measure estimates and obtain coincidence of the free boundaries. We also derive Liouville type theorems for entire solutions. These results are new even for linear systems.

Fully nonlinear dead-core systems

Abstract

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system and obtain higher sharp regularity across the free boundary. Additionally, we prove geometric measure estimates and obtain coincidence of the free boundaries. We also derive Liouville type theorems for entire solutions. These results are new even for linear systems.

Paper Structure

This paper contains 11 sections, 12 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existence of solutions and weak comparison principle
  3. Existence of solutions
  4. Weak comparison principle
  5. Regularity of solutions at the free boundary
  6. Geometric properties of the free boundary
  7. Non-degeneracy of solutions
  8. Porosity of the free boundary
  9. Coincidence of the free boundaries
  10. Liouville type results
  11. Radial analysis

Key Result

Theorem 2.1

If $T:X \to X$, where $X$ is a Banach space, is continuous and compact, and the set is bounded, then $T$ has a fixed point.

Figures (2)

  • Figure 1: The growth of $|(u,v)|$ near the free boundary
  • Figure 2: Radial solution

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • ...and 14 more