Fine structure of rupture set for semilinear elliptic equation with singular nonlinearity

TL;DR

This paper analyzes the structure of rupture sets for semilinear elliptic equations with singular nonlinearities, establishing rectifiability, sharp estimates, and improved regularity results, with applications to evolution equations.

Contribution

It provides a detailed geometric and measure-theoretic analysis of rupture sets, including rectifiability and stratification, and improves regularity estimates for solutions.

Findings

Rupture set is (n-2)-rectifiable with sharp Minkowski content estimates.

Stratification based on tangent function symmetry leads to k-rectifiability.

Enhanced integrability of derivatives under boundedness assumptions.

Abstract

In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ \Delta u=u^{-p}+f,\,\,u\geq 0\text{ in }\Omega\subset\mathbb{R}^n, $$ where $ n\geq 2 $, $ p>1 $, $ \Omega $ is a bounded domain, and $ f\in L^q(\Omega) $ with $ \frac{1}{2}+\frac{1}{2p}<\frac{q}{n} $. We establish a sharp estimate for the Minkowski content of the rupture set $ \{u=0\} $ and demonstrate that this set is $ (n-2) $-rectifiable. For this, we examine the stratification of the rupture set based on the symmetry properties of tangent functions, leading to the proof of $ k $-rectifiability for each $ k $-stratum. As a significant byproduct of our analysis, we improve the integrability of $ D^ju $ with $ j\in\mathbb{Z}_+ $ to the optimal Lorentz space $ L^{\frac{2(p+1)}{j(p+1)-2},\infty} $, under the assumption that $ D^{j-1}f $ is bounded. As an application of our results in the static case of the equation, for a class of suitable weak solutions to the three-dimensional evolutional problem $$ \partial_tu=\Delta u-u^{-p},\,\,u\geq 0\text{ in }(\Omega\subset\mathbb{R}^3)\times(0,T), $$ where $ p>3 $ and $ T>0 $, we show that $ \{u(\cdot,t)=0\} $ is $ 1 $-rectifiable for a.e. $ t\in(0,T) $.