Mapping theorems for Sobolev spaces of vector-valued functions

Abstract

We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces X = {0} and Y , each Lipschitz continuous mappingBut 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 (Ω, X).

Sobolev Gradients for the Möbius Energy

“…Proof. We follow the argumentation in Theorem 5.1 from [1]: The norm ψ := • X is Lipschitz with Lipschitz constant 1. Utilizing the Sobolev embedding for scalar functions (Theorem A.2) and for s ∈ ]0, 1] along with the chain rule (Theorem A.1), we obtain…”

“…Let γ : T → R m be a sufficiently smooth embedding 1 of the circle T into Euclidean space. Its Möbius energy [27,61] is defined as (1) where γ (x, y) denotes the length of the shortest arc of γ connecting γ (x) and γ (y).…”

“…R.), and by DFG-Project 282535003: Geometric curvature functionals: energy landscape and discrete methods (both authors). 1 In many cases we consider curves γ being differentiable a.e. but not necessarily C 1 .…”

On gradient flows with obstacles and Euler’s elastica

“…The argument will use difference quotient techniques, which are also available for vector-valued functions. For a detailed overview about these methods see [7].…”

Degenerate equations in a diffusion–precipitation model for clogging porous media

“…Therefore, sincec is assumed to belong to L 2 (0, T; H 1 0 ( )), it is reasonable that also ∇θ can be estimated in L 2 (0, T; L 2 ( )). For this, we test (3.1) with ∇θ which leads to 1 2 ∇θ (t) 2 2…”