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.

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.

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.

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 $[0,\,T]$ .

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.

Regularity of the Very Weak Solutions for Double Obstacle Problems

2010 ◽  
Vol 143-144 ◽  
pp. 1396-1400
Author(s):  
Xu Juan Xu ◽  
Xiao Na Lu ◽  
Yu Xia Tong
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Minkowski Inequality ◽  
Young Inequality ◽  
Obstacle Problems ◽  
Very Weak Solutions ◽  
Local Integrability ◽  
Regularity Results ◽  
Harmonic Equation ◽  
Homogeneous Elliptic Equation

The apply of harmonic eauation is know to all. Our interesting is to get the regularity os their solution, recently for obstacle problems. Many interesting results have been obtained for the solutions of harmonic equation ande their obstacle problems, however the double obstacle problems about the definition and regularity results for non-homogeneous elliptic equation. In this paper , the basic tool for the Young inequality,Hölder inequality, Minkowski inequality, Poincaré inequality and a basic inequality. The definition of very weak solutions for double obstacle problems associated with non-homogeneous elliptic equation is given, and the local integrability result is obtained by using the technique of Hodge decomposition.

Sign in / Sign up

Export Citation Format

Share Document