Abstract
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
Highlights
The present paper deals with elliptic equations in planar domains with mixed boundary conditions and aims at proving asymptotic expansions and unique continuation properties for solutions near boundary points where a transition from Dirichlet to Neumann boundary conditions occurs.A great attention has been devoted to the problem of unique continuation for solutions to partial differential equations starting from the paper by Carleman [5], whose approach was based on some weighted a priori inequalities
An alternative approach to unique continuation was developed by Garofalo and Lin [14] for elliptic equations in divergence form with variable coefficients, via local doubling properties and Almgren monotonicity formula; we quote [18] for quantitative uniqueness obtained by monotonicity methods
The method based on doubling properties and Almgren monotonicity formula has been successfully applied to treat the problem of unique continuation from the boundary in [1, 2, 9, 19, 27] under homogeneous Dirichlet conditions and in [26] under homogeneous Neumann conditions
Summary
The present paper deals with elliptic equations in planar domains with mixed boundary conditions and aims at proving asymptotic expansions and unique continuation properties for solutions near boundary points where a transition from Dirichlet to Neumann boundary conditions occurs. We extend the procedure developed in [9,10,11,12,13] to the case of mixed Dirichlet/Neumann boundary conditions, providing sharp asymptotic estimates for solutions near the Dirichlet-Neumann junction and, as a consequence, unique continuation properties. The proof of Theorem 2.1 is based on the study of the monotonicity properties of the Almgren function N and on a fine blow-up analysis which will be performed in Sections 4 and 5
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