Boundary Regularity in the Sobolev Imbedding Theorems

1966 ◽  
Vol 18 ◽  
pp. 350-356
Author(s):  
A. E. Hurd
Keyword(s):  
Partial Differential Equations ◽  
Euclidean Space ◽  
Differential Equations ◽  
Boundary Regularity ◽  
Function Spaces ◽  
Negative Integer ◽  
Partial Differential ◽  
Image Position

In (6) (see also 7), Sobolev introduced a class of function spaces Wm,p(Ω) (m a non-negative integer, 1 < p < ∞) defined on open subsets Ω of Euclidean space En, which have important applications in partial differential equations. They are defined as follows. For each n-tuple α = (α1, … αn) of non-negative integers let

XVI.—Simultaneous Linear Partial Differential Equations of the Second Order

1943 ◽  
Vol 61 (3) ◽  
pp. 195-209
Author(s):  
E. L. Ince
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Second Order ◽  
Partial Differential ◽  
Compatible System ◽  
Linear Partial Differential Equations ◽  
Polynomial Coefficients ◽  
Linearly Independent ◽  
Image Position

SummaryIn the system of two linear partial differential equations of the second ordera,…,f were supposed to be polynomials in x, and a1…, f1 polynomials in y. These polynomial coefficients were subjected to certain restrictions, including conditions for the system having exactly four linearly independent solutions, and conditions for preserving the symmetrical aspect, in x and y, of the system. It has been proved that any compatible system of the contemplated form whose coefficients satisfy the stipulated conditions is equivalent with, i.e. transformable into, a hypergeometric system. More particularly it has been shown that the hypergeometric systems involved are the system of partial differential equations associated with Appell's hypergeometric function in two variables F2 and the confluent systems arising herefrom.

The Hardy Space H1 on Manifolds and Submanifolds

1972 ◽  
Vol 24 (5) ◽  
pp. 915-925 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Partial Differential Equations ◽  
Hardy Space ◽  
Euclidean Space ◽  
Differential Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Riesz Transforms ◽  
Partial Differential ◽  
Integrable Functions

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables

1965 ◽  
Vol 17 ◽  
pp. 676-686 ◽  
Author(s):  
R. P. Gilbert
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Complex Variable ◽  
Entire Functions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Singular Behaviour ◽  
Image Position

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation(1.1)where A (r2), C(r2) are entire functions of the complex variable

On Some Relations Between Partial and Ordinary Differential Equations

1958 ◽  
Vol 10 ◽  
pp. 183-190 ◽  
Author(s):  
Erwin Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Operators ◽  
Basic Tool ◽  
Partial Differential ◽  
Image Position

The theory of solutions of partial differential equations (1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).

scholarly journals Function Spaces with a Random Variable Exponent

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Boping Tian ◽  
Yongqiang Fu ◽  
Bochi Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Variable Exponent ◽  
Random Variable ◽  
Function Spaces ◽  
Partial Differential

The spaces with a random variable exponent and are introduced. After discussing the properties of the spaces and , we give an application of these spaces to the stochastic partial differential equations with random variable growth.

scholarly journals On Self-Adjoint Partial Differential Equations of the Second Order

1924 ◽  
Vol 43 ◽  
pp. 35-38 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Expression ◽  
Second Order ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Differential ◽  
Image Position

Let be a linear differential expression involving n independent variables xi the coefficients AikBi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

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