**Title: **Symmetry properties of entire solutions to some elliptic PDEs and phase transition

**Abstract: **
In the first part of the seminar I will discuss some symmetry
properties of solutions to some semilinear elliptic PDEs. We consider solutions
decaying only in some variables and we look for sufficient conditions for
them to be radially symmetric in those variables (joint work with Alberto
Farina and Andrea Malchiodi).

In the second part of the talk I will give an idea of the work that I am doing
now with Andrea Malchiodi about phase transition. The aim is to show that any
nondegenerate compact Willmore surface embedded in $R^3$ can be obtained as a limit
of nodal sets of solution of the weighted Cahn-Hilliard equation.

This seminar is part of the AJS series of seminars.