scholarly journals Integral representation and relaxation of local functionals on Cheeger–Sobolev spaces

2022 ◽  
Vol 217 ◽  
pp. 112744
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Integral Representation ◽  
Sobolev Spaces

scholarly journals Integral representation of local functionals depending on vector fields

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Fares Essebei ◽  
Andrea Pinamonti ◽  
Simone Verzellesi
Keyword(s):  
Integral Representation ◽  
Sobolev Spaces ◽  
Vector Fields ◽  
First Order ◽  
Bounded Set

Abstract Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

scholarly journals On the relaxation of variational integrals in metric Sobolev spaces

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Integral Representation ◽  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
General Framework ◽  
Metric Measure Spaces ◽  
Representation Theorems ◽  
Differentiable Structure ◽  
Convex Case ◽  
Measure Spaces ◽  
Variational Integrals

AbstractWe give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for “relaxed” variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger–Keith's differentiable structure.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

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