AI 中文总结
本文针对边界条件受小Robin扰动的二阶椭圆型偏微分方程模型的电压势，推导了一般渐近公式，并分析了新定义的Robin-Dirichlet容量、Robin-Neumann容量与经典度量的比较关系。
AI 中文摘要
受反问题和形状优化相关问题的启发，本文推导了电压势的渐近展开式，该电压势是二阶椭圆型偏微分方程模型的解，其边界条件受到小扰动。更具体地说，在问题的固定参考构型中，齐次Dirichlet或齐次Neumann边界条件被Robin边界条件取代，该Robin边界条件在环境域边界的“小”子集ω_ε上具有导纳k_ε>0，且当ε→0时该导纳趋于零。在上述两种情形中，本文针对电压势建立了一般渐近公式，该公式依赖于关于消失子集ω_ε的形状和参数k_ε的最少假设。这些展开式的尺度（用于衡量扰动强度）由称为“Robin-Dirichlet”容量或“Robin-Neumann”容量的新量来衡量，这些量取决于ω_ε的几何形状和参数k_ε的值。本文分析了这些量如何根据k_ε在ε→0时的行为，与ω_ε“小”度的更经典度量（如ω_ε的容量或Neumann容量）进行比较。
英文摘要
Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance $k_\varepsilon > 0$ on a ``small'' subset $ω_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset $ω_\varepsilon$ and the parameter $k_\varepsilon$. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called ``Robin-Dirichlet'' capacity or a ``Robin-Neumann'' capacity, which depend on the geometry of $ω_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the ``smallness'' of $ω_\varepsilon$, such as the capacity or the Neumann capacity of $ω_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.
Comments38 pages, 1 figure
