Local compactness and nonvanishing for weakly singular nonlocal quadratic forms

Abstract

In this work we study a class of nonlocal quadratic forms given by Ej(u,v)=12∫RN∫RN(u(x)−u(y))(v(x)−v(y))j(x−y)dxdy,where j:RN→[0,∞] is a measurable even function with min{1,|⋅|2}j∈L1(RN). Assuming merely j∉L1(RN), we show local compactness of the embedding Dj(RN)↪L2(RN), where Dj(RN) denotes the space of functions u∈L2(RN) with Ej(u,u)<∞. Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in Dj(RN). As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in Dj(RN). Our main results extend to cylindrical unbounded sets of the type Ω=U×Rk, where U⊂RN−k is open and bounded. Finally, we note that a Poincaré inequality associated with Ej holds for unbounded domains of this type, thereby extending the corresponding result in Felsinger et al. (2015) for bounded domains.