Geometry and analysis seminar: Xavier Ros-Oton

Speaker: Xavier Ros-Oton University of Texas at Austin

Title: Pohozaev identities for integro-differential operators

Abstract: The Pohozaev identity satisfied by solutions of \(-\Delta u = f(x,u)\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\), is a classical result which has many applications. For instance, it leads to sharp nonexistence results, to uniqueness of solutions, monotonicity formulas, and to symmetry properties. Moreover, the identity is also used in other contexts such as the wave equation or harmonic maps. The aim of this talk is to present new Pohozaev identities for the fractional Laplacian and other integro-differential operators. Quite surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over \(\partial\Omega\)) which is completely local. More precisely, in these new identities the function \(u/d^s|_{\partial\Omega}\) plays the role that the normal derivative \(\partial u/\partial\nu\) plays in the classical one, where \(d(x) = {\rm dist}(x,\partial\Omega)\).