A quantitative Talenti-type comparison result with Robin boundary conditions
Vincenzo Amato, Rosa Barbato, Simone Cito, Alba Lia Masiello, Gloria Paoli
TL;DR
The paper formulates and proves a quantitative Talenti-type comparison for the Poisson equation with Robin boundary conditions, linking the solution on a Lipschitz domain $\Omega$ to the rearranged problem on the ball $\Omega^\sharp$ via the Schwarz rearrangement and the datum $f^\sharp$. Using a detailed level-set analysis and propagation of asymmetry, it proves a rigidity result (equality implies $\Omega$ is a ball and $u,f$ are radially symmetric up to translation) and introduces stability estimates in Lorentz norms: for suitable $k$, the gaps $\|v\|_{L^{k,1}(\Omega^\sharp)}-\|u\|_{L^{k,1}(\Omega)}$ and $\|v\|_{L^{2k,2}(\Omega^\sharp)}^2-\|u\|_{L^{2k,2}(\Omega)}^2$ are bounded below by constants times $\alpha(\Omega)^2$, with explicit dependence on $|\Omega|$, $\|f\|_1$, $\beta$, and the isoperimetric constant $\gamma_n$. In 2D with $f\equiv1$, a pointwise bound yields $\|v-u^\sharp\|_{L^\infty(\Omega^\sharp)} \ge C_3(\Omega)\alpha(\Omega)^3$, and a $\beta$-robust form extends toward Dirichlet behavior as $\beta\to\infty$. As applications, the results provide alternative proofs of quantitative Saint-Venant (Robin torsion) and, in planar domains, Bossel–Daners (first Robin eigenvalue) inequalities, highlighting the interconnection between Robin and Dirichlet settings. The work thus advances the understanding of stability and rigidity in Talenti-type isoperimetric comparisons under Robin boundary conditions and opens questions about optimal exponents and norm refinements.
Abstract
The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions of a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.