$C^{1,\mathrm{Dini}}$ 域中带奇异势的椭圆方程奇异集的界
Bounds of singular sets for elliptic equations in $C^{1, Dini}$ domains with singular potentials
- Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究带奇异势的椭圆方程在 $C^{1,\mathrm{Dini}}$ 域边界邻域中奇异集的定量余维二估计，通过加倍指标上界和边界展平获得显式界。
AI中文摘要：
我们研究在边界邻域内方程 $\Delta u+V(x)u=0$（在 $\Omega$ 内）且 $u=0$（在 $partial\Omega$ 上）的解的奇异集的定量余维二估计，其中 $\Omega\subset\mathbb R^n$ 是有界 $C^{1,\mathrm{Dini}}$ 域，且 $V\in L^p(\Omega)$ 对某个 $p>n$ 成立。我们首先证明加倍指标的显式上界由 $C(n,p,\Omega)(1+\\|V\\|_{L^p(\Omega)}^{\frac{2p}{3p-2n}})$ 给出。解析输入是对于具有一致椭圆 Dini 主系数和 $V\in L^p$ 的二阶椭圆方程解的内部体积估计。利用定量加倍指标界和边界展平，我们给出了 $C^{1,\mathrm{Dini}}$ 域边界邻域内奇异集的显式上界。
英文摘要：
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.
