Pohozaev-like identity for the regional fractional laplacian

We establish a new integration by parts formula for the regional fractional laplacian $(-\Delta)^s_\Omega$ 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.