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

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 methods in combination with regularity results for viscosity solutions, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.