Robin boundary conditions in shape optimization

The present thesis is concerned with the problem of proving the existence of optimal domains for functionals subjected to Robin Boundary conditions. We treat both cases of positive and negative Robin parameters. In the case of positive Robin parameters we prove the existence of a minimizing domain in a class of Lipschitz domains of given measure, that are uniform extension domains. In addition to the linear case, i.e. the case of the first eigenvalue, we consider Rayleigh quotients corresponding to the Sobolev embedding theorem, up to the critical exponent. Subsequently, we show that the volume constraint can be replaced by a surface area constraint.For negative Robin parameters we restrict the class of domains. We consider domains that are starshaped with respect to a fixed ball, thus fixing the topology of the domains. This exludes recent counter examples to the reverse Faber-Krahn inequality. Using a uniform trace inequality, we prove the existence of a maximizing domain for the first eigenvalue of the Robin Laplacian. Subsequently, we present an additional existence result in a class resembling spherical shells. Moreover, we prove the existence of optimal domains in a smoother setting, using a constraint on the mean curvature to obtain the compactness of the class of domains with respect to the stronger topology. As a consequence of the smoother setting, we are able to discuss further regularity properties of optimal domains