Università di Napoli Frederico II

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.
In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.
The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).
 

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations  with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets  that are  $L^1$-close.  As an application, we address the global and local minimality of certain lamellar configurations.