scholarly journals Integrability of Very Weak Solutions for Boundary Value Problems of Nonhomogeneous p-Harmonic Equations

2019 ◽  
pp. 1-9
Author(s):  
Yeqing Zhu ◽  
Yanxia Zhou ◽  
Yuxia Tong
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Boundary Value ◽  
Very Weak Solutions ◽  
Very Weak Solution ◽  
Harmonic Equation

The paper deals with very weak solutions u to boundary value problems of the nonhomogeneous p-harmonic equation. We show that, any very weak solution u to the boundary value problem is integrable provided that r is sufficiently close to p.

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials

1989 ◽  
Vol 113 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
Nicolas Charalambakis ◽  
François Murat
Keyword(s):  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Dependent ◽  
Initial Boundary Value

SynopsisWe prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

scholarly journals The Behaviour of Weak Solutions of Boundary Value Problems for Linear Elliptic Second Order Equations in Unbounded Cone-Like Domains

2016 ◽  
Vol 30 (1) ◽  
pp. 203-217
Author(s):  
Damian Wiśniewski
Keyword(s):  
Weak Solution ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Boundary Value ◽  
Continuity Condition ◽  
Second Order ◽  
Second Order Equations ◽  
Elliptic Divergence

AbstractWe investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.

scholarly journals Weak solutions in Elasticity of dipolar bodies with stretch

2013 ◽  
Vol 29 (1) ◽  
pp. 33-40
Author(s):  
MARIN MARIN ◽  
◽  
GABRIEL STAN ◽  
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Linear Operators ◽  
Elastic Materials ◽  
Elastic Bodies ◽  
Dipolar Bodies

In the present paper we generalize the results obtained by Iesan and Quintanilla for microstretch elastic bodies in order to cover the dipolar elastic materials with stretch. For the boundary value problem considered in this context, we use some results from the theory of semigroups of the linear operators in order to prove the existence and uniqueness of a weak solution.

REGULARITY OF VERY WEAK SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATION

2013 ◽  
Vol 15 (04) ◽  
pp. 1350012 ◽  
Author(s):  
WEI ZHANG ◽  
JIGUANG BAO
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Weak Solutions ◽  
Local Regularity ◽  
Difference Quotient ◽  
Very Weak Solutions ◽  
Very Weak Solution ◽  
Bootstrap Argument ◽  
The Difference ◽  
Higher Regularity

In this paper, we study the local regularity of very weak solution [Formula: see text] of the elliptic equation Dj(aij(x)Diu) = f - Digi. Using the bootstrap argument and the difference quotient method, we obtain that if [Formula: see text], [Formula: see text] and [Formula: see text] with 1 < p < ∞, then [Formula: see text]. Furthermore, we consider the higher regularity of u.

Very weak solutions of boundary value problems for the Laplace operator and the Lamé system on polyhedral domains in ℝ3

1994 ◽  
Vol 124 (4) ◽  
pp. 645-672
Author(s):  
Ding Hua
Keyword(s):  
Boundary Value Problems ◽  
Laplace Operator ◽  
Weak Solutions ◽  
Boundary Value ◽  
Very Weak Solutions ◽  
Nonsmooth Data ◽  
Lamé System ◽  
Variational Solutions ◽  
Lame System ◽  
The Laplace Operator

The notion of very weak solutions is introduced in this paper in order to solve the boundary value problems for the Laplace operator and for the Lamé system with nonsmooth data in polyhedral domains. A continuity theorem is given for variational solutions of the above problems. This result may be used to solve problems with concentrated loads.

Local Boundedness for Very Weak Solutions of Leray-Lions Equation

2012 ◽  
Vol 457-458 ◽  
pp. 210-213
Author(s):  
Jian Tao Gu ◽  
Chun Xia Gao ◽  
Yu Xia Tong
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Decomposition Methods ◽  
Hodge Decomposition ◽  
Very Weak Solutions ◽  
Local Boundedness ◽  
Very Weak Solution

The local boundedness of very weak solution of Leray-Lions equation is given in this paper by Hodge decomposition methods.

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