Optimal partitions for Robin Laplacian eigenvalues

TL;DR

This paper establishes the existence of optimal partitions that minimize the sum of Robin Laplacian eigenvalues across multiple disjoint open sets with rectifiable boundaries within a specified domain.

Contribution

It proves the existence of optimal partitions for a multiphase shape optimization problem involving Robin Laplacian eigenvalues, extending previous results to more general boundary conditions.

Findings

Existence of optimal partitions for Robin Laplacian eigenvalues.

Optimal partitions have rectifiable boundaries.

Framework applicable to general domains in R^d.

Abstract

We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of $k$ mutually disjoint {\it open} sets which have a $\mathcal H ^ {d-1}$-countably rectifiable boundary and are contained into a given box $D$ in $R^d$