A surgery result for the spectrum of the Dirichlet Laplacian

In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy. As well, we apply this result to prove existence of solutions for shape optimization problems of spectral type with both measure and perimeter constraints.