Radial Symmetry of Minimizers for Some Variational Problems

We discuss the radial symmetry of minimizers of variational problems like 

 ∫_RN |∇u|² dx + ∫_RN F(u) dx 

under the constraint  ∫_RN G(u) dx = c.

We also consider some modifications of it (the integrals are in a radially symmetric domain (ball, annulus, the exterior of a ball), F(r,u) and G(r,u) depend on the space variable in a radial way, problems without constraint.) We make a comparison among the following methods:

• Schwarz symmetrization;

• Gidas, Ni and Nirenberg;

• Reflection method;

Besides more classical results, we present recent results for nonlocal problems obtained in a joint work with M. Maris (Besan ̧con).