标题： 关于诺伊曼-泊松算子和自伴传输问题 显示英文标题

标题： On Neumann-Poincaré operators and self-adjoint transmission problems

摘要： 我们讨论在$L^2$环境下，作为$-\nabla\cdot h\nabla$作用的算子，其中分段常数函数$h$沿 Lipschitz 超曲面$\Sigma$具有跳跃，而无需对$h$的符号做出显式假设。 我们建立了若干充分条件，以确保具有$H^s$正则性的算子在适当$s\in[1,\frac{3}{2}]$的情况下为自伴算子，这些条件基于跳跃值、正则性和$\Sigma$的几何特性。一个重要的中间步骤是与 Lipschitz 设定下$\Sigma$上 Neumann-Poincaré 算子的 Fredholm 性质之间的联系，这是新的。 显示英文摘要

摘要： We discuss the self-adjointness in $L^2$-setting of the operators acting as $-\nabla\cdot h\nabla$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $\Sigma$, without explicit assumptions on the sign of $h$. We establish a number of sufficient conditions for the self-adjointness of the operator with $H^s$-regularity for suitable $s\in[1,\frac{3}{2}]$, in terms of the jump value and the regularity and geometric properties of $\Sigma$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincar\'e operator on $\Sigma$, which is new for the Lipschitz setting.