Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian

TL;DR

This paper investigates the asymptotic behavior of free boundaries in the exterior Bernoulli problem involving the half Laplacian, revealing how the free boundary interacts with the convex envelope of the fixed boundary under various conditions.

Contribution

It provides new insights into the qualitative properties and asymptotic behavior of free boundaries in the half Laplacian Bernoulli problem, especially as the gradient parameter varies.

Findings

Free boundary behavior as the gradient parameter approaches zero or infinity.

Perpendicular rays of the free boundary meet the convex envelope of the fixed boundary.

Conditions under which the free boundary interacts with the convex envelope.

Abstract

In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to $0^+$ and to $+\infty$. Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meet the convex envelope of the fixed boundary.