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On the Laplace operator and on the vector potential problems in the half-space: an approach using weighted spaces. (English) Zbl 1290.35034
Summary: The purpose of the present paper is twofold. The first object is to study the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half-space of \(\mathbb{R}^n\). The behaviour of solutions at infinity is described by means of a family of weighted Sobolev spaces. A class of existence, uniqueness and regularity results are obtained. The second purpose is to investigate some properties of grad, div and curl operators in order to treat curl-div systems of the form \(\mathrm{curl}w = u\), \(\mathrm{div} w = 0\) and problems related to vector potentials and Helmholtz decomposition.

MSC:
35E20 General theory of PDEs and systems of PDEs with constant coefficients
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35F15 Boundary value problems for linear first-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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