Regularity of very weak solutions for elliptic equation of divergence form

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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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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Bounded very weak solutions for some non-uniformly elliptic equation with L^1 datum

Annales Academiae Scientiarum Fennicae Mathematica ◽

10.5186/aasfm.2017.4205 ◽

2017 ◽

Vol 42 ◽

pp. 95-103

Author(s):

Chao Zhang ◽

Shulin Zhou

Keyword(s):

Elliptic Equation ◽

Weak Solutions ◽

Very Weak Solutions

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Regularity of the Very Weak Solutions for Double Obstacle Problems

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.143-144.1396 ◽

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.

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$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Boundary Value Problems ◽

10.1186/s13661-020-01480-8 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hui Wang ◽

Caisheng Chen

Keyword(s):

Parabolic Equation ◽

Weak Solutions ◽

Nonlinear Parabolic Equation ◽

Divergence Form ◽

Decay Estimates ◽

Laplacian Equation ◽

Estimates Of Solutions ◽

Nonlinear Parabolic ◽

Doubly Nonlinear ◽

Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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