We introduce a new family of interpolation inequalities: in dimension $d\geq 2$, $$\left(\int_{\mathbb{R}^d}\frac{|w|^{1+p}}{|x|^{\gamma}}\,dx\right)^{\frac1{1+p}}\leq \mathsf C_{\beta,\gamma,p}\;\left(\int_{\mathbb{R}^d}\frac{|\nabla w|^{2}}{|x|^{\beta}}\,dx\right)^{\frac{\vartheta}{2}}\left(\int_{\mathbb{R}^d}\frac{|w|^{2p}}{|x|^{\gamma}}\,dx\right)^{\frac{1-\vartheta}{2p}}\;\;\forall w\in C^{\infty}_c(\mathbb{R}^d),$$ valid in the range of parameters $p\in (0,1)$, $\gamma\in(-\infty,d)$, $\gamma-2<\beta<\tfrac{d-2}{d}\gamma$, and where the exponent $\vartheta$ is determined by the scaling invariance.

These inequalities 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 interpolation 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 $\mathsf C_{\beta,\gamma,p}$. Although the Euler-Lagrange equations are invariant under rotation, we prove that the extremal functions are not radially symmetric, provided $\gamma$ and $\beta$ are chosen appropriately. Our proof is variational and relies on a linear stability analysis of radially symmetric solutions. 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) and Matteo Muratori (Politecnico di Milano). This work is partially supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (ANR-10-LABX-0098, LabEx SMP) and partially supported by the Project EFI (ANR-17-CE40-0030).