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

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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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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