Local Boundedness for Very Weak Solutions of Leray-Lions Equation

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.

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

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2019/v32i330147 ◽

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.

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REGULARITY OF VERY WEAK SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATION

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500120 ◽

2013 ◽

Vol 15 (04) ◽

pp. 1350012 ◽

Cited By ~ 3

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.

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

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-020-00646-8 ◽

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 ) .

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.69 ◽

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 $[0,\,T]$ .

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