Mapping theorems for Sobolev-spaces of vector-valued functions

TL;DR

This paper investigates the conditions under which Lipschitz and Gateaux differentiable mappings induce bounded operators between Sobolev spaces of vector-valued functions, revealing the role of the Radon-Nikodym Property and duality in these mappings.

Contribution

It establishes new mapping theorems for Sobolev spaces with Banach space values, including conditions for Lipschitz and Gateaux differentiable functions, and applies these to embedding theorems and compactness results.

Findings

Lipschitz mappings induce bounded operators iff the target space has the Radon-Nikodym Property.

Gateaux differentiability ensures operator boundedness without additional conditions.

Results lead to new embedding theorems and a multi-dimensional Aubin-Lions Lemma.

Abstract

We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces $X\neq\{0\}$ and $Y$, each Lipschitz continuous mapping $F:X\rightarrow Y$ gives rise to a mapping $u\mapsto F\circ u$ from $W^{1,p}(\Omega,X)$ to $W^{1,p}(\Omega,Y)$ if and only if $Y$ has the Radon-Nikodym Property. But if $F$ 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 $W^{1,p}_0(\Omega,X)$.