In this paper we prove the existence of an optimal set for the minimization of the $k$th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schrödinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.

中文翻译：

---

与符号变化电容测量相关的 p-Laplacian 较高特征值的优化结果

在本文中，我们证明了最小化 $克$的变分特征值 $磷$- 拉普拉斯算子 $磷$- 包含在有限测度盒中的固定测度的准开集。的特征值得到了类似的存在结果$磷$-与薛定谔势相关的拉普拉斯算子。为了处理这些非线性形状优化问题，我们开发了一种通用方法，可以处理特征值的连续相关性$磷$- 拉普拉斯算子与符号改变容量措施相关 $\gamma$-收敛。