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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.

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
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