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

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. $t\in(0,T)$.