  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.
