Bounds of singular sets for elliptic equations in $C^{1, Dini}$ domains with singular potentials

Abstract

We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \begin{equation*} Δu+V(x)u=0\qquad\text{in }Ω, \qquad u=0\qquad\text{on }\partialΩ, \end{equation*} where $Ω\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(Ω)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,Ω)(1+\|V\|_{L^p(Ω)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.