We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set Ω with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as Ω. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.

中文翻译：

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具有混合边界条件的椭圆问题解的精确估计

我们表明，使用对称化技术，可以证明比较原理（我们主要关注 ${升}^1$比较）在平滑有界集 Ω 上的椭圆偏微分方程的解与相当一般的边界条件，与定义在与 Ω 相同体积的球上的合适相关问题的解之间。这包括例如混合问题，其中 Dirichlet 边界条件规定在部分边界上，而 Robin 边界条件规定在其补上。