On the regularity of very weak solutions for linear elliptic equations in divergence form
2020 ◽
Vol 27
(5)
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Keyword(s):
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 ) .
2013 ◽
Vol 15
(04)
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pp. 1350012
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2012 ◽
Vol 457-458
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pp. 863-866
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1983 ◽
Vol 27
(1)
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pp. 1-30
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2012 ◽
Vol 262
(4)
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pp. 1867-1878
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2000 ◽
Vol 23
(5)
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pp. 313-318
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2012 ◽
Vol 457-458
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pp. 210-213
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