We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green’s functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics.

中文翻译：

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凸等势面的两个不等式

我们建立了两个几何不等式，分别针对外部Dirichlet问题中的调和函数和针对内部Dirichlet问题中的格林函数（边界表面光滑且凸）建立了两个几何不等式。这两个不等式都涉及等势面上平均曲率和高斯曲率的积分，以及谐波势的正态导数。这些不等式概括了二维空间中等电位曲线的几何守恒律，并为三维静电中的两个自由边界问题提供了解决方案。