We investigate the continuity and differentiability of the Hardy constant with respect to perturbations of the domain in the case where the problem involves the distance from a boundary submanifold. We also investigate the case where only the submanifold is deformed.

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$.

We consider a class of eigenvalue problems for polyharmonic operators, includingDirichlet and buckling-type eigenvalue problems. We prove an analyticity result for thedependence of the symmetric functions of the eigenvalues upon domain perturbations andcompute Hadamard-type formulas for the Frechét differentials. We also considerisovolumetric domain perturbations and characterize the corresponding critical domains forthe symmetric functions of the eigenvalues. Finally, we prove that balls are criticaldomains.