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The Neumann problem for elliptic equations with non-smooth coefficients. (English) Zbl 0807.35030
The so-called Neumann $$(N_ p)$$ and regularity $$(R_ p)$$ problems for a divergence form symmetric elliptic operator $$L = \text{div} A \nabla$$ are considered. Here $$A = (a_{ij} (x))$$ is a matrix of bounded measurable functions. The problem is considered in the unit ball $$B$$, but the methods of proof are sufficiently general to allow $$B$$ to be a bounded starlike Lipschitz domain.
The results in this work are of two types. First, the meaning of and consequences of solvability of the Neumann and regularity problems for general operator $$L$$ are considered. A Neumann function for $$L$$ is introduced. It is proved that the same sort of estimates known for the Green’s function of $$L$$ are satisfied. This leads to a potential representation for solutions of the Neumann problem and to uniqueness results for weak solutions. Further nontangential convergence results for the Neumann and regularity problem with data in $$L^ p (\partial B)$$ are considered. These results are utilized to investigate the general consequences of solvability of $$(N_ p)$$ and $$(R_ p)$$. An example which show that the Neumann problem for data in $$L^ p (d \sigma)$$ need not be solvable for any arbitrary elliptic operator is given.
The results of the second type concern solvability of $$(N_ p)$$ and $$(R_ p)$$ for certain classes of operators. Specifically, if $$A(\theta)$$ is an elliptic symmetric matrix $$a_{ij} \equiv a_{ij} (\theta)$$, $$\theta \in S^{n-1}$$ (radial independence) then $$(N_ 2)$$ and $$(R_ 2)$$ are solvable for $$L = \text{div} A \nabla$$ in the unit ball $$B$$. And if $$A(r, \theta)$$ is a small perturbation of $$A(\theta) = A(1, \theta)$$ then $$(N_ 2)$$ and $$(R_ 2)$$ are solvable for $$L = \text{div} A(r, \theta) \nabla$$. At the end of the paper the authors indicate a list of several questions which remain open and some directions for further research.

##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J67 Boundary values of solutions to elliptic equations and elliptic systems
##### Keywords:
non-smooth coefficients; Green’s function
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##### References:
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