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I am gathering material that contains results of regularity for solutions of the equation \begin{equation} \mbox{div}(A(x) \nabla u) = f \end{equation} where the coefficints $a_{i,j}$ of $A$ are only measurable and $f \in L^p$ for some $p \in \mathbb{N}$ and \begin{equation} \lambda |\xi|^{2} \le \langle A(x)\xi , \xi \rangle \le |\xi|^{2} \quad \forall x,\xi \end{equation}. Can you give me some references? Assume others hypothesis if you want. Thank you.

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The paper "Besov regularity of elliptic boundary value problems" by Dahlke and Devore is quite insightful in the case where A=const but f and the shape of the domain boundary are rough. –  Nick Alger Jul 2 '12 at 23:00

1 Answer 1

up vote 1 down vote accepted
  1. Ladyzhenskaya & Uraltseva, "Linear and quasilinear elliptic equations"
  2. Gilbarg & Trudinger, "Elliptic PDE of second order"
  3. Maly & Ziemer, "Fine regularity of solutions of elliptic PDE"
  4. Heinonen, Kilpelainen, Martio "Nonlinear potential theory of degenerate elliptic equations"

Listed in approximate order of relevance to the question. The books 3 & 4 deal more with p-Laplace type equations, but you can still get something by setting $p=2$.

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