Variational Integrals on Orlicz-Sobolev Spaces

1998 ◽  
Vol 17 (2) ◽  
pp. 393-415 ◽  
Author(s):  
M. Fuchs ◽  
V. Osmolovski
Keyword(s):  
Sobolev Spaces ◽  
Variational Integrals

scholarly journals Γ-convergence of nonconvex integrals in Cheeger--Sobolev spaces and homogenization

2017 ◽  
Vol 10 (4) ◽  
pp. 381-405 ◽  
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variational Integrals ◽  
Γ Convergence

AbstractWe study Γ-convergence of nonconvex variational integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces. Applications to relaxation and homogenization are given.

The gap phenomenon for variational integrals in Sobolev spaces

1992 ◽  
Vol 120 (1-2) ◽  
pp. 93-98 ◽  
Author(s):  
M. Giaquinta ◽  
G. Modica ◽  
J. Souček
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Spaces ◽  
Variational Integrals

SynopsisWe show that a gap phenomenon occurs for general variational integrals for mappings from a domain Rn into a Riemannian manifold if has a non-trivial topology.

scholarly journals A new proof of semicontinuity by Young measures and an approximation theorem in Orlicz-Sobolev spaces

2003 ◽  
Vol 2003 (15) ◽  
pp. 881-898
Author(s):  
Barbara Bianconi
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Approximation Theorem ◽  
Growth Conditions ◽  
Young Measures ◽  
New Approach ◽  
General Growth ◽  
Variational Integrals

We give a new approach to study the lower semicontinuity properties of nonautonomous variational integrals whose energy densities satisfy general growth conditions. We apply the theory of Young measures and properties of Orlicz-Sobolev spaces to prove semicontinuity result.

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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