Symmetry breaking in weighted interpolation inequalities: the porous medium regime

We introduce a new family of interpolation inequalities, with two radial power-law weights and exponents in the subcritical range. They are related with the so-called entropy- entropy production inequalities in the problem of intermediate asymptotics for nonlinear diffusions, and play a role for the porous medium equation similar to some standard Caffarelli-Kohn-Nirenberg inequalities for the fast diffusion equation.

We address the question of symmetry breaking: are the extremal functions radially symmetric or not? By extremal functions we mean functions that realize the equality case in the inequality, written with optimal constants. Although the Euler-Lagrange equations are invariant under rotation, we prove that the extremal functions are not radially symmetric, provided the power laws of the weights are chosen appropriately. Our proof of the symmetry breaking is variational and relies on the stability analysis of optimal solutions in the class of radially symmetric functions. The core of the proof consists of finding the optimal constant in a weighted Hardy-Poincaré inequality.

This work is a collaboration with Jean Dolbeault (CEREMADE, Université Paris Dauphine) and Matteo Muratori (Politecnico di Milano).