A glimpse on regularity theory: from PDEs to interfaces

We consider elliptic equations in the double-divergence form and perform a two-fold analysis. First, we present results on the improved regularity of the solutions along zero level-sets. In this context, solutions become asymptotically Lipschitz; under further conditions on the data of the problem, we produce asymptotically-Lipschitz regularity for the gradient of the solutions. A second instance of analysis regards the geometric properties of nodal sets; our findings concern the (local) regularity and the Hausdorff dimension of those sets. We close the talk with a discussion on applications to solid mechanics, as well as with comments on further directions of research.