Mapping theorems for Sobolev spaces of vector-valued functions
Volume 240 / 2018
Studia Mathematica 240 (2018), 275-299
MSC: Primary 46E40; Secondary 46E35.
DOI: 10.4064/sm8757-4-2017
Published online: 11 September 2017
Abstract
We consider Sobolev spaces with values in Banach spaces, with emphasis on mapping properties. Our main results are the following: Given two Banach spaces $X\not =\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u \mapsto F\circ u$ from $W^{1,p}(\varOmega ,X)$ to $W^{1,p}(\varOmega ,Y)$ if and only if $Y$ has the Radon–Nikodým Property. But if in addition $F$ is one-sided Gateaux differentiable, no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin–Lions Lemma and characterizations of the space $W^{1,p}_0(\varOmega ,X)$.
