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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ domain and $V\in L^p(\mathrm{\Omega })$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,\mathrm{\Omega })(1+\mathrm{\parallel }V{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^{\frac{2p}{3p-2n}})$.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 $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ domain.