On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval

TL;DR

This paper investigates the existence and multiplicity of solutions to the interior Bernoulli free boundary problem involving the fractional Laplacian on an interval, identifying a critical Bernoulli constant and exploring solution properties.

Contribution

It establishes the existence of a critical Bernoulli constant for fractional Laplacian problems on intervals and analyzes solution multiplicity and variational connections.

Findings

Existence of a critical Bernoulli constant $\lambda_{ ext{alpha,D}}\

Multiple solutions exist depending on the parameter $\lambda\

Solutions for $\\alpha=1$ can be non-minimizers of the variational problem.

Abstract

We study the structure of solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval $D$ with parameter $\lambda > 0$. In particular, we show that there exists a constant $\lambda_{\alpha,D} > 0$ (called the Bernoulli constant) such that the problem has no solution for $\lambda \in (0,\lambda_{\alpha,D})$, at least one solution for $\lambda = \lambda_{\alpha,D}$ and at least two solutions for $\lambda > \lambda_{\alpha,D}$. We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for $\alpha = 1$ that there exist solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval which are not minimizers of the corresponding variational problem.