Improved regularity for a nonlocal dead-core problem

TL;DR

This work studies a nonlocal two-phase dead-core problem driven by the fractional Laplacian $-(-\Delta)^s$ with $s\in(1/2,1)$ and a reaction term $u_+^\gamma - u_-^\gamma$ for $\gamma\in(0,1/3)$. The authors develop a scaling growth analysis combined with Liouville-type results to obtain sharp regularity at branching points, where the solution and certain derivatives vanish, without relying on the maximum principle. The main result shows that near a branching point $x_0$, the solution satisfies $|u(x)| \le C\|u\|_{L^\infty}|x-x_0|^{\frac{2s}{1-\gamma}}$, hence $u\in C^{\frac{2s}{1-\gamma}}$ at such points, with a transition index $\nu\in\{1,2\}$ determined by $s$ and $\gamma$. The approach extends to fully nonlinear operators, allows passing to the local limit $s\to1$, and yields consequences for local two-phase problems, gradient estimates, and potential extensions to nonlinear elliptic regimes, thereby bridging nonlocal and local two-phase theories.

Abstract

We obtain improved regularity results for solutions to a nonlocal dead-core problem at branching points. Our approach, which does not rely on the maximum principle, introduces a new strategy for analyzing two-phase problems within the local framework, an area that remains largely unexplored.

Theorems & Definitions (29)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Definition 4.1