Sobolev Spaces and L 2 Regularity Theory

2012 ◽  
pp. 255-309
Author(s):  
Jürgen Jost
Keyword(s):  
Sobolev Spaces ◽  
Regularity Theory

scholarly journals Regularity for a fractional p-Laplace equation

2017 ◽  
Vol 20 (01) ◽  
pp. 1750003 ◽  
Author(s):  
Armin Schikorra ◽  
Tien-Tsan Shieh ◽  
Daniel E. Spector
Keyword(s):  
Laplace Operator ◽  
Laplace Equation ◽  
Sobolev Spaces ◽  
Homogeneous Equation ◽  
Regularity Theory ◽  
Complex Interpolation ◽  
Natural Analogue ◽  
Classical Formula

In this note, we consider regularity theory for a fractional [Formula: see text]-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the [Formula: see text]-Laplacian. We obtain the natural analogue to the classical [Formula: see text]-Laplacian situation, namely [Formula: see text]-regularity for the homogeneous equation.

scholarly journals Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems with Rough Coefficients

2012 ◽  
Vol 64 (6) ◽  
pp. 1395-1414 ◽  
Author(s):  
Scott Rodney
Keyword(s):  
Functional Analysis ◽  
Quadratic Form ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Principal Part ◽  
Regularity Theory ◽  
Existence Theory ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Rough Coefficients

Abstract This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the formThe principal part ξ'P(x)ξ of the above equation is assumed to be comparable to a quadratic form Q(x,ξ)=ξ'Q(x)ξ that may vanish for non-zero ξ ∊ ℝn. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ)=W1,2(Θ,Q) and QH10(Θ)= W1,20 (Θ,Q)as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains

1997 ◽  
Vol 127 (1) ◽  
pp. 77-125 ◽  
Author(s):  
Benqi Guo ◽  
Ivo Babuška
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Elliptic Problems ◽  
Regularity Theory ◽  
Weighted Spaces ◽  
Regularity Of Solutions ◽  
The Finite Element Method ◽  
Normed Spaces ◽  
Nonsmooth Domains

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

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.

Sign in / Sign up

Export Citation Format

Share Document