Minimality and stability properties in Sobolev and isoperimetric inequalities

This thesis addresses the characterization of minimizers in various Sobolev- and isoperimetric-type inequalities and the analysis of the corresponding stability phenomena. We first investigate a family of variational problems which arise in connection to a suitable interpolation between the classical Sobolev and Sobolev trace inequalities. We provide a full characterization of minimizers for each problem, in turn deriving a new family of sharp constrained Sobolev inequalities on the half-space. We then prove novel stability results for the Sobolev inequality and for the anisotropic isoperimetric inequality. Both of these results share the feature of being “strong-form” stability results, in the sense that the deficit in the inequality is shown to control the strongest possible distance to the family of equality cases.