Gradient Flows for Dirichlet Eigenvalues

We are interested in existence of gradient flows for shape functionals especially for first Laplacian eigenvalues. We introduce different techniques to prove existence and use different formulations for gradient flows. We apply a compactness argument to prove existence of generalized minimizing movements for Dirichlet and Robin boundary conditions with respect to several common metrics. Moreover, we use Brunn-Minkowski inequalities to prove $\alpha$-convexity and existence of contraction semi-groups for Dirichlet boundary conditions on convex bodies. Finally, we give a proof of $\alpha$-convexity for Robin boundary conditions by the second domain variation since we do not have a Brunn-Minkowski inequality in this case.