标题： 关于粗糙积分元的散度定理中允许内边界的最优类域 显示英文标题

标题： An optimal class of domains permitting inner boundaries in a divergence theorem for rough integrands

摘要： 研究了有限测度的开区域$U \subset \mathbb{R}^m$上粗糙类$\mathcal{DM}_\infty(U)$中有有限Radon测度散度的有界向量场$u \, \colon \, U \to \mathbb{R}^m$的散度定理。设$\mathcal{DB}(U)$表示在Lebesgue空间$L_\infty(U)$中$\mathcal{DM}_\infty(U)$的闭包。 证明了域 $U$ 满足散度定理的等价条件 \begin{equation} \exists ! \ell \in \mathcal{DB}'(U) \, \colon \, \int_U \, \mathrm{d} \, \mathrm{div} u = \ell(u) \quad \forall u \in \mathcal{DM}_\infty(U), \end{equation}，其中泛函 $\ell$ 通过作为集中在域边界的有限可加测度的等价类来推广经典的曲面测度。 即，以下与这个散度定理等价：(a) 散度积分$$ \mathcal{DM}_\infty(U) \to \mathbb{R} \, \colon \, u \mapsto \int_U \, \mathrm{d} \, \mathrm{div} u $$在一致拓扑下连续，(b) 非外边界$\partial U \setminus \mathrm{ext}_* U$的 1-余维 Hausdorff 测度有限，其中$\partial U$是拓扑边界，$\mathrm{ext}_* U$是测度论上的外部，(c) 集合指示函数$\chi_U$是从下方向由梯度在$L_1$中一致有界的光滑函数序列的极限，(d) 集合$U$是从内部由具有均匀有界周长的光滑集合序列的极限。 这提供了一个最优的正则性类的区域，在这些区域内，关于$\mathcal{DM}_\infty$-场的散度定理成立，并且具有在一致拓扑下连续的曲面泛函。 与经典的表述不同，新的表述可以考虑到被积函数沿1维内边界上的不连续性。 显示英文摘要

摘要： The divergence theorem on an open domain $U \subset \mathbb{R}^m$ of finite measure is investigated for the rough class $\mathcal{DM}_\infty(U)$ of bounded vector fields $u \, \colon \, U \to \mathbb{R}^m$ with a finite Radon measure divergence. Let $\mathcal{DB}(U)$ denote the closure of $\mathcal{DM}_\infty(U)$ in the Lebesgue space $L_\infty(U)$. Equivalent conditions are proved for the domain $U$ to admit the divergence theorem \begin{equation} \exists ! \ell \in \mathcal{DB}'(U) \, \colon \, \int_U \, \mathrm{d} \, \mathrm{div} u = \ell(u) \quad \forall u \in \mathcal{DM}_\infty(U), \end{equation} where the functional $\ell$ generalizes classical surface measure by being an equivalence class of finitely additive measures concentrated on the domain boundary. Namely, the following are equivalent to this divergence theorem: (a) the divergence integral $$ \mathcal{DM}_\infty(U) \to \mathbb{R} \, \colon \, u \mapsto \int_U \, \mathrm{d} \, \mathrm{div} u $$ is continuous in the uniform topology, (b) the non-exterior boundary $\partial U \setminus \mathrm{ext}_* U$ has finite 1-codimensional Hausdorff measure, where $\partial U$ is the topological boundary, and $\mathrm{ext}_* U$ is the measure theoretic exterior, (c) the set indicator function $\chi_U$ is a (sequential) limit from below of smooth functions with gradient uniformly bounded in $L_1$, (d) the set $U$ is a (sequential) limit from inside of smooth sets with uniformly bounded perimeter. This provides an optimal regularity class of domains on which the divergence theorem for $\mathcal{DM}_\infty$-fields holds with a surface functional that is continuous in the uniform topology. Unlike the classical formulation, the new formulation can take discontinuities of the integrand along 1-codimensional inner boundaries into account.