We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

中文翻译：

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具有非局部竞争项的谱形优化问题

我们研究了作为 Dirichlet Laplacian 的第一特征值和 Riesz 型相互作用泛函的相对强度之和的谱泛函的最小化。我们表明，当Riesz排斥强度低于临界值时，会出现最小化剂。然后，通过展开分析证明，当Riesz推力足够小时，球是刚性最小化器。最终我们证明，对于 Riesz 排斥的某些机制，规则的极小值不存在。