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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.
##### MSC:
 35J25 Boundary value problems for second-order elliptic equations
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