Abstract
In this paper we consider a linear elliptic equation in divergence form 0.1∑i,jDj(aij(x)Diu)=0inΩ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \\quad \\hbox {in } \\Omega . \\end{aligned}$$\\end{document}Assuming the coefficients a_{ij} in W^{1,n}(Omega ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution uin L^{n'}_mathrm{loc}(Omega ) of (0.1) is actually a weak solution in W^{1,2}_mathrm{loc}(Omega ).
Highlights
Let n ≥ 2 and Ω ⊂ Rn be a bounded open set
In this paper we study regularity properties of very weak solutions to the linear elliptic equation
Let u be a very weak solution of (1.1), with A(x) = (aij(x))i,j satisfying (1.2), (1.3) and (1.4), u belongs to Wl1o,c2(Ω) and it is a weak solution
Summary
Let n ≥ 2 and Ω ⊂ Rn be a bounded open set. In this paper we study regularity properties of very weak solutions to the linear elliptic equation. Where the matrix-field A : Ω → Rn×n, A(x) = (aij(x))i,j, is elliptic and belongs to W 1,n(Ω, Rn×n) ∩ L∞(Ω, Rn×n), i.e. for some positive constants λ, Λ, and M. The matrix A is symmetric, that is aij = aji a.e. in Ω for all i, j ∈ {1, ..., n}
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