On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

TL;DR

This paper investigates Bernoulli free boundary problems involving the half Laplacian and spectral half Laplacian, establishing geometric properties of solutions and inequalities under certain assumptions.

Contribution

It provides new results on the shape and properties of solutions, including starshapedness and convexity, using variational and Beurling methods.

Findings

Starshapedness of solutions under starshaped data

Convexity of solutions under convex data

Brunn-Minkowski inequality for the Bernoulli constant

Abstract

We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational approach, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.