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Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form
Annales de l'Institut Henri Poincaré C, Analyse non linéaire ( IF 1.8 ) Pub Date : 2020-11-27 , DOI: 10.1016/j.anihpc.2020.11.002 Baptiste Trey
中文翻译:
某些涉及散度形式算子特征值的函数的最优集的正则性
更新日期:2020-11-27
Annales de l'Institut Henri Poincaré C, Analyse non linéaire ( IF 1.8 ) Pub Date : 2020-11-27 , DOI: 10.1016/j.anihpc.2020.11.002 Baptiste Trey
In this paper we consider minimizers of the functional where is a bounded open set and where are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets have finite perimeter and that their free boundary is composed of a regular part, which is locally the graph of a -regular function, and a singular part, which is empty if , discrete if and of Hausdorff dimension at most if , for some .
中文翻译:
某些涉及散度形式算子特征值的函数的最优集的正则性
在本文中,我们考虑了泛函的极小值 在哪里 是一个有界开集,其中 是具有 Dirichlet 边界条件和 Hölder 连续系数的散度形式的算子 Ω 上的前k 个特征值。我们证明最优集合 具有有限周长并且它们的自由边界 由一个规则部分组成,它是一个局部图-regular 函数,以及一个单数部分,如果是空的, 离散如果 和至多豪斯多夫维数 如果 , 对于一些 .