Abstract:
For any (possibly unbounded) simply connected domain $\Omega\subset\mathbb{R}^2$ whose complement has nonempty interior, we establish an explicit relation between the solving operators of the elliptic Dirichlet and Neumann boundary-value problems for classes of weak solutions with first derivatives from $L_p(\Omega)$. It is assumed that the uniformly elliptic operators are of divergence form with essentially bounded matrix coefficients and with functionals on the right-hand side which are bounded on the spaces of corresponding weak solutions. The relation between the solving operators is established under the necessary and sufficient solvability conditions for the Neumann problem, i.e., when the given functional does vanish on the subspace of constants.
Keywords:elliptic equation, divergence form, general form of a linear continuous functional, essentially bounded matrix-valued coefficients, solving operator, first-order systems, Douglis–Nirenberg ellipticity, weak solution, Dirichlet problem, Neumann problem.
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