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Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

Galiano Gonzalo

Opuscula Mathematica

2024

Vol. 44, no. 5

707--726

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti’s theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti’s proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz’s rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti’s theorem.