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.

scholarly journals Sobolev spaces on Lie groups: Embedding theorems and algebra properties

2019 ◽  
Vol 276 (10) ◽  
pp. 3014-3050 ◽  
Author(s):  
Tommaso Bruno ◽  
Marco M. Peloso ◽  
Anita Tabacco ◽  
Maria Vallarino
Keyword(s):  
Lie Groups ◽  
Sobolev Spaces ◽  
Embedding Theorems ◽  
And Algebra

On a class of weighted Sobolev spaces and the minimization of quadratic forms

1978 ◽  
Vol 81 (1-2) ◽  
pp. 119-130
Author(s):  
Frans Penning ◽  
Niko Sauer
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Compact Support ◽  
Quadratic Forms ◽  
Weighted Sobolev Spaces ◽  
Weight Functions ◽  
Compact Embedding ◽  
Embedding Theorems ◽  
Smooth Functions ◽  
Square Integrability

SynopsisIn this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the spaceRnor a half-space ofRn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

Weighted Sobolev spaces and exterior problems for the Helmholtz equation

1987 ◽  
Vol 410 (1839) ◽  
pp. 373-383 ◽  
Keyword(s):  
Hilbert Space ◽  
Helmholtz Equation ◽  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radiation Condition ◽  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Uniqueness Of Solutions ◽  
Exterior Problems

Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.

scholarly journals Lifting of Quaternionic Frames to Higher Dimensions with Partial Ridges

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Florian Heinrich ◽  
Brigitte Forster
Keyword(s):  
Sobolev Spaces ◽  
Higher Dimensions ◽  
Embedding Theorems ◽  
Wavelet Frames ◽  
Skew Field ◽  
Novel Approach

AbstractWe consider the technique of lifting frames to higher dimensions with the ridge idea that originally was introduced by Grafakos and Sansing. We pursue a novel approach with regard to a non-commutative setting, concretely the skew-field of quaternions. Moreover, we allow for splitting dimensions and for lifting with regard to multi-ridges. To this end, we introduce quaternionic Sobolev spaces and prove the corresponding embedding theorems. We mention as concrete examples quaternionic wavelet frames and quaternionic shearlet frames, and give the respective lifted families.

Approximation numbers of embeddings of Sobolev spaces

1996 ◽  
Vol 221 (1) ◽  
pp. 177-187
Author(s):  
D. E. Edmunds ◽  
A. A. Ilyin
Keyword(s):  
Sobolev Spaces ◽  
Approximation Numbers

Approximation numbers of embeddings of Sobolev spaces

1996 ◽  
Vol 221 (2) ◽  
pp. 177-187
Author(s):  
D. E. Edmunds ◽  
A. A. Ilyin
Keyword(s):  
Sobolev Spaces ◽  
Approximation Numbers

Differentiability of the Energy Functional of a Class of Quasi-Linear Elliptic

2014 ◽  
Vol 484-485 ◽  
pp. 1033-1037
Author(s):  
Xiao Li Pan ◽  
Li Hua Mu ◽  
Hui Chen
Keyword(s):  
Sobolev Spaces ◽  
Convergence Theorem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Energy Functional ◽  
Embedding Theorem ◽  
Variation Principle ◽  
Embedding Theorems ◽  
Generalized Sobolev Spaces ◽  
Dominated Convergence

To study the differentiability of a class of quasi-linear elliptic equations energy functional by using the embedding theorems and some other properties of generalized Sobolev spaces. According to the variation principle the energy functional of equations is expressed in some appropriate generalized Sobolev spaces, the various parts of differentiability of the energy functional of equations are discussed under certain conditions. Using Lebesgue dominated convergence theorem and embedding theorem, the energy functional of equations is proved. Finally, the overall differentiability is proved. The conclusions lay the foundation for the next step to prove the existence of the critical point of the energy functional, that is, the existence of solutions of the equations.

scholarly journals Compact embedding theorems and a Lions' type Lemma for fractional Orlicz–Sobolev spaces

2021 ◽  
Vol 300 ◽  
pp. 487-512
Author(s):  
Edcarlos D. Silva ◽  
M.L. Carvalho ◽  
J.C. de Albuquerque ◽  
Sabri Bahrouni
Keyword(s):  
Sobolev Spaces ◽  
Compact Embedding ◽  
Embedding Theorems
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