# Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian

## TL;DR

This paper establishes a new uniqueness theorem for an obstacle problem involving the fractional Laplacian, extending classical results and employing probabilistic potential theory applicable to various integro-differential operators.

## Contribution

It introduces a novel proof technique based on probabilistic potential theory for the fractional Laplacian obstacle problem, broadening applicability beyond classical operators.

## Key findings

- Proved uniqueness for the obstacle problem with fractional Laplacian
- Extended classical Laplace operator results to fractional case
- Developed a new probabilistic proof method

## Abstract

We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-differential operators.