The study of geometric properties of solutions of Partial Differential Equations is of great interest. One of the most celebrated results in this topic is the one obtained by B. Gidas, W. M. Ni and L. Nirenberg in [1], where the authors establish radial symmetry to nonnegative solutions of the problem \[
\begin{cases}-\Delta u=f(u) & \text { on } B \\ u=0 & \text { on } \partial B\end{cases} \] where $B \subset \mathbb{R}^N$ (with $N \geqslant 2$) is a ball. Later on, several authors went on to generalize this result in several directions.

In this talk, I present a quantitative version of a Gidas-Ni-Nirenberg-type symmetry result involving the $p$-Laplacian. Quantitative stability is achieved here via integral identities based on the proof of rigidity established by J. Serra in 2013, which extended to general dimension and the $p$-Laplacian operator an argument proposed by P. L. Lions in dimension 2 for the classical Laplacian.

In passing, we obtain a quantitative estimate for the measure of the singular set and an explicit uniform gradient bound.

This work was done in collaboration with S. Dipierro, G. Poggesi, E. Valdinoci.

References

1. B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209-243.