Uniqueness for Very Weak Solution to a Class of Elliptic Equations

2012 ◽  
Vol 457-458 ◽  
pp. 863-866
Author(s):  
Xu Juan Xu ◽  
Jian Tao Gu ◽  
Xiao Li Liu
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Hodge Decomposition ◽  
Very Weak Solutions ◽  
Uniqueness Property ◽  
Very Weak Solution

This paper studies on the very weak solution to a class of elliptic equations , and acquire the uniqueness property for very weak solutions by means of the Hodge decomposition and others.

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.

scholarly journals On the regularity of very weak solutions for linear elliptic equations in divergence form

2020 ◽  
Vol 27 (5) ◽  
Author(s):  
Domenico Angelo La Manna ◽  
Chiara Leone ◽  
Roberta Schiattarella
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Continuity Condition ◽  
Very Weak Solutions ◽  
Linear Elliptic Equation ◽  
Very Weak Solution

Abstract In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

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.

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.

On the energy equality for very weak solutions to 3D MHD equations

2021 ◽  
pp. 1-24
Author(s):  
Baishun Lai ◽  
Yifan Yang
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Weak Solutions ◽  
Mhd Equations ◽  
Time Interval ◽  
Very Weak Solutions ◽  
Energy Equality ◽  
Very Weak Solution

In this paper, we consider the energy equality of the 3D Cauchy problem for the magneto-hydrodynamics (MHD) equations. We show that if a very weak solution of MHD equations belongs to $L^{4}(0,\,T;L^{4}(\mathbb {R}^{3}))$ , then it is actually in the Leray–Hopf class and therefore must satisfy the energy equality in the time interval $[0,\,T]$ .

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

Uniqueness for Very Weak Solution to a Class of Elliptic Equations

2012 ◽  
Vol 457-458 ◽  
pp. 863-866
Author(s):  
Xu Juan Xu ◽  
Jian Tao Gu ◽  
Xiao Li Liu
Keyword(s):  
Weak Solution ◽  
Elliptic Equations ◽  
Very Weak Solution

Qualitative Analysis of Solutions for a Class of Anisotropic Elliptic Equations with Variable Exponent

2015 ◽  
Vol 59 (3) ◽  
pp. 541-557 ◽  
Author(s):  
G. A. Afrouzi ◽  
M. Mirzapour ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Special Form ◽  
Variable Exponent ◽  
Nonlinear Term ◽  
Fountain Theorem ◽  
Image Position ◽  
Anisotropic Elliptic Equations

AbstractWe are concerned with the degenerate anisotropic problemWe first establish the existence of an unbounded sequence of weak solutions. We also obtain the existence of a non-trivial weak solution if the nonlinear termfhas a special form. The proofs rely on the fountain theorem and Ekeland's variational principle.

scholarly journals Very weak solutions to elliptic equations with singular convection term

2018 ◽  
Vol 457 (2) ◽  
pp. 1376-1387 ◽  
Author(s):  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Convection Term ◽  
Very Weak Solutions
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