Quantitative stratification and sharp regularity estimates for supercritical semilinear elliptic equations

TL;DR

This paper develops a detailed stratification theory for singular sets of supercritical elliptic equations, providing sharp regularity estimates and Hausdorff dimension bounds for solutions and their derivatives.

Contribution

It introduces a stratification framework for singularities in supercritical elliptic equations and derives sharp interior regularity estimates using Reifenberg-type theorems.

Findings

Hausdorff dimension of stratified singular sets is at most k

Stratified sets are k-rectifiable

Solutions exhibit sharp Lorentz space regularity estimates for derivatives

Abstract

In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$ -\Delta u=|u|^{p-1}u\quad\text{in }\Omega,\quad p>\frac{n+2}{n-2}, $$ where $ \Omega\subset\mathbb{R}^n $ is a bounded domain with $ n\geq 3 $. Our primary focus is on the structure of stratification for the singular sets. We define the $ k $-th stratification $ S^k(u) $ of $ u $ based on the tangent functions and measures. We show that the Hausdorff dimension of $ S^k(u) $ is at most $ k $ and $ S^k(u) $ is $ k $-rectifiable, and establish estimates for volumes associated with points that have lower bounds on the regular scales. These estimates enable us to derive sharp interior estimates for the solutions. Specifically, if $ \alpha_p=\frac{2(p+1)}{p-1} $ is not an integer, then for any $ j\in\mathbb{Z}_{\geq 0} $, we have $$ D^ju\in L_{\operatorname{loc}}^{q_j,\infty}(\Omega), $$ which implies that for any $ \Omega'\subset\subset\Omega $, $$ \sup\{\lambda>0:\lambda^{q_j}\mathcal{L}^n(\{x\in\Omega':|D^ju(x)|>\lambda\})\}<+\infty, $$ where $ \mathcal{L}^n(\cdot) $ is the $ n $-dimensional Lebesgue measure, and $$ q_j=\frac{(p-1)(\lfloor\alpha_p\rfloor+1)}{2+j(p-1)}, $$ with $ \lfloor\alpha_p\rfloor $ being the integer part of $ \alpha_p $. The proofs of these results rely on Reifenberg-type theorems developed by A. Naber and D. Valtorta to study the stratification of harmonic maps.