Pohozaev-like identity for the regional fractional laplacian

Abstract

We establish a new integration by parts formula for the regional fractional laplacian $(-\Delta {)}_{\Omega }^s$ in bounded open sets of class $C^2$. As a direct application, we prove that weak solutions to the corresponding Dirichlet problem satisfy a Pohozaev-like identity with an explicit remainder term. We apply the later to eigenvalue problems in the unit ball and discuss its potential use in establishing boundary-type unique continuation properties.

The identity presented in the article is likely incorrect because the argument used to show that the remainder term in the identity is well defined is not valid. The proof relies on the claimed fact (stated in the last paragraph of page 473 of [1]) that weak solutions belong to (H^{2s-1/2}). However, this regularity statement is false. This issue was brought to our attention by Xavier Ros-Oton