# Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

## TL;DR

This paper rigorously characterizes the fractional Laplacian with nonlocal exterior conditions using forms method, showing the associated semigroups are submarkovian, ultracontractive, and bounded between Dirichlet and Neumann semigroups.

## Contribution

It provides a new form-based characterization of fractional Laplacian operators with nonlocal exterior conditions and analyzes their semigroup properties.

## Key findings

- Operators generate strongly continuous submarkovian semigroups.
- Semigroups are ultracontractive.
- Robin semigroup is bounded between Dirichlet and Neumann semigroups.

## Abstract

Let $\Omega\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms, we give a rigorous characterization of the realization in $L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$) with the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical local case $s=1$, it turns out that the nonlocal (Robin and Neumann) exterior conditions are incorporated in the form domain. We show that each of the above operators generates a strongly continuous submarkovian semigroup which is also ultracontractive. In the second part, we show that the semigroup corresponding to the nonlocal Robin exterior condition is always sandwiched between the fractional Dirichlet semigroup and the fractional Neumann semigroup.