Local compactness and nonvanishing for weakly singular nonlocal quadratic forms. (April 2020) - Jarohs, Sven; Weth, Tobias
Abstract: In this work we study a class of nonlocal quadratic forms given by E j ( u, v ) = 1 2 ∫ R N ∫ R N ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) j ( x − y ) d x d y, where j : R N → [ 0, ∞ ] is a measurable even function with min 1, | ⋅ | 2 j ∈ L 1 ( R N ) . Assuming merely j ∉ L 1 ( R N ), we show local compactness of the embedding D j ( R N ) ↪ L 2 ( R N ), where D j ( R N ) denotes the space of functions u ∈ L 2 ( R N ) with E j ( u, u ) < ∞ . Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in D j ( R N ) . As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in D j ( R N ) . Our main results extend to cylindrical unbounded sets of the type Ω = U × R k, where U ⊂ R N − k is open and bounded. Finally, we note that a Poincaré inequality associated with E j holds for unbounded domains of this type, thereby extending the corresponding result in Felsinger et al. (2015) for bounded domains.