Isoperimetric inequalities for eigenvalues by direct methods

A possible way to prove isoperimetric inequalities may be:

- prove the existence of an optimal shape, without imposing any regularity constraint on the admissible shapes. A relaxation phenomenon may be observed, e.g. the solution is a measure or a quasi-open set.

- prove the regularity of the optimal shape. This problem is often very difficult, but sometimes a mild regularity result is enough in order to pass to the next step.

- extract optimality conditions and get extra information about the optimal shape. If the regularity is not very high, classical optimality conditions obtained by shape derivative methods can not be written. Other optimality criteria have to be used.

This plan will be discussed for the minimization of the the eigenvalues of the Laplacian with Dirichlet and Robin boundary conditions.

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