具有相反符号Robin边界条件的椭圆型问题的区域摄动
Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign
摘要：
本文考虑了Robin边值条件下挠问题的能量，当解不是极小值点时。它的依赖关系的域的体积和边界的表面积进行了讨论。与正弹性常数的情况相反，球不提供最小值。对于近似球形的区域和弹性常数接近于零的能量是最大的球。这个结果是真实的一般域在平面上的第一个非平凡的Steklov特征值的一个附加条件。对于更一般的弹性常数的情况是更多的参与，它是密切相关的特定域扰动。本文所用的方法是解的级数表示，即用Steklov本征函数、一阶和二阶形状导数以及Payne和Weinberger关于扭转刚度的等周不等式\cite{PaWe61}表示。
英文摘要：
We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive elasticity constants, the ball does not provide a minimum. For nearly spherical domains and elasticity constants close to zero the energy is largest for the ball. This result is true for general domains in the plane under an additional condition on the first non-trivial Steklov eigenvalue. For more general elasticity constants the situation is more involved and it is strongly related to the particular domain perturbation. The methods used in this paper are the series representation of the solution in terms of Steklov eigenfunctions, the first and second shape derivatives and an isoperimetric inequality of Payne and Weinberger \cite{PaWe61} for the torsional rigidity.
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