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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.

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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[1] Hanouzet, Rendiconti del Seminario Matemetico della Università di Padova XLVI pp 227– (1971)
[2] Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thèse de Doctorat d’Etat, Université Pierre et Marie Curie, Paris, 1987.
[3] Amrouche, Journal de Mathématiques Pures et Appliquées 73 pp 576– (1994)
[4] Amrouche, Journal de Mathematiques Pures et Appliquees 76 pp 55– (1997)
[5] Girault, Journal of Faculty of Science, University of Tokyo Sect IA 39 pp 279– (1992)
[6] The Stokes Problem and Vector Potential Operator in Three-Dimensional Exterior Domains. Publications du Laboratoire d’Analyse Numérique, R. 92026: UPMC, Paris, 1992.
[7] von Wahl, Mathematical Methods in the Applied Sciences 15 pp 123– (1992)
[8] Bolik, Mathematical Methods in the Applied Sciences 20 pp 737– (1997)
[9] Problèmes aux limites non homogènes et applications. Dunod: Paris, 1968.
[10] Barros Neto, Ann. Scuola Norm. Sup. Pisa 19 pp 331– (1965)
[11] Weighted Sobolev Spaces. Wiley: Chichester, 1985. · Zbl 0601.46039
[12] Hardy-type Inequalities. Wiley: New York, 1985.
[13] Galdi, Archive for Rational Mechanics and Analysis 112 pp 291– (1990)
[14] Inequalities. Cambridge University Press: Cambridge, 1952. · Zbl 0047.05303
[15] Etude des champs de Beltrami dans des domaines de ?3 bornés et non-bornés et applications en astrophysique. Thèse de Doctorat, Laboratoire d’analyse numérique, Université Pierre et Marie Curie, Paris (1999).
[16] Variétés différentiables. Hermann: Paris, 1960.
[17] Babu?ka, Numerische Mathematik 20 pp 179– (1973)
[18] Brezzi, Revue Française d’Automatique, Informatique et Recherche Opérationnelle Série Rouge. Analyse Numérique R2 pp 129– (1974)
[19] Elliptic Problems in Non-Smooth Domains. Pitman: London, 1985.
[20] Sobolev Spaces. Academic press: New York, 1975.
[21] Benci, Annali di Matematica Pure ed Applicata 121 pp 319– (1979)
[22] Weighted Sobolev spaces for the Laplace equation in the half-space. In Comptes Rendus de l’Académie des Sciences, Série I, Paris, accepted.
[23] Borchers, Hokkaido Mathematical Journal 19 pp 67– (1990) · Zbl 0719.35014
[24] Chaljub-Simon, Journal of Differential Equations 76 pp 374– (1988) · Zbl 0636.34026
[25] An Introduction to the Mathematical Study of Navier-Stokes Equations, vol. I-II. Springer: Berlin, 1994.
[26] Finite Element Methods for Navier-Stokes Equations. Springer: Berlin, 1986. · Zbl 0585.65077
[27] Gobert, Journal of Mathematical Analysis and Applications 36 pp 518– (1971)
[28] Héron, Communication in Partial Differential Equations 6 pp 1301– (1981)
[29] The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach: New York, 1964.
[30] Luczy?ski, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques XIV pp 381– (1966)
[31] Muckenhoupt, Transactions of American Mathematical Society 165 pp 207– (1972)
[32] Les méthodes directes en théorie des équations elliptiques. Masson: Paris, 1967.
[33] Equations aux dérivées partielles. Presse de l’université de Montréal: Montréal, 1966.
[34] Noussair, Journal of Differential Equations 57 pp 349– (1985)
[35] Piat, Journal of Convex Analysis 1 pp 135– (1994)
[36] McOwen, Communications in Pure and Applied Mathematics 32 pp 783– (1979)
[37] Siegel, Pacific Journal of Mathematics 175 pp 571– (1996) · Zbl 0865.35038
[38] A new approach to the Helmholtz decomposition and Neumann problem in Lq-spaces for bounded and exterior domains. In Mathematical Problems Relating to the Navier-Stokes Equation. (editor). World Scientific: Singapore, 1992; 1-35.
[39] Zhou, Complex Variables 33 pp 1– (1993) · Zbl 0796.47037
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