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Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains. (English) Zbl 1414.35067
The paper deals with mixed boundary value problems for the elliptic operator \[ L=-\sum_{i,j=1}^n a_{ij}(x)D_iD_j \] on unbounded domains \(\mathcal{D}\subset\mathbb{R}^n\), \(n\geq 3,\) with Dirichlet boundary condition specified on a part of \(\partial\mathcal{D}\) and Neumann-type condition given on the rest of the boundary. The coefficients matrix \(\{a_{ij}\}\) is assumed to be only positive semidefinite \(\mathcal{D}\), although it is required to be positive definite on certain finite layers of \(\mathcal{D}\).
The main result proved is a Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of \(\partial\mathcal{D}\) and the “thickness” of its Dirichlet portion. The result obtained is formulated in terms of the \(s\)-capacity of the Dirichlet portion \(\partial\mathcal{D}\), while the Neumann boundary should satisfy certain “admissibility” condition in the sequence of layers converging to infinity.
35J25 Boundary value problems for second-order elliptic equations
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