Embeddings for A-weakly differentiable functions on domains

By Franz Gmeineder, Bogdan Raiță. Journal of Functional Analysis, 2017

58 citations recorded by the source corpus.

Abstract We prove that the inhomogeneous estimate of vector fields on balls in R n ( ∫ B | D k − 1 u | n / ( n − 1 ) d x ) ( n − 1 ) / n ⩽ c ( ∫ B | A u | + | u | d x )  for all  u ∈ C ∞ ( B ¯ , R N ) holds if and only if the linear, constant coefficient differential operator A of order k has finite dimensional null-space (FDN). This generalizes the Gagliardo–Nirenberg–Sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full-space ( ∫ R n | D k − 1 u | n / ( n − 1 ) d x ) ( n − 1 ) / n ⩽ c ∫ R n | A u | d x  for all  u ∈ C c ∞ ( R n , R N ) , proved by Van Schaftingen precisely for elliptic and cancelling (EC) operators, building on fundamental L 1 -estimates from the works of Bourgain and Brezis. We prove that FDN strictly implies EC and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/e9b9fa90f7c1660776a3c4fd0bf83a40c85d8bd7