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Dirichlet and Neumann exterior problems for the \(n\)-dimensional Laplace operator. An approach in weighted Sobolev spaces. (English) Zbl 0878.35029
The authors present a solution to Dirichlet and Neumann boundary value problems for the Laplace equation on an exterior domain in \(\mathbb{R}^n\). Instead of imposing explicit asymptotic conditions at infinity they require that the solution be from an appropriate weighted Sobolev space with power-logarithmic weights. This appears advantageous for both the theoretical and numerical approach because it enables to control not only the behaviour of the gradient but also of the solution itself. The boundary of the domain is considered to be Lipschitz-continuous if \(p=2\) and of the class \(\mathcal C^{1,1}\) if \(p\neq2\).

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. on pure and applied math., Vol. XII, 623-727, (1959) · Zbl 0093.10401
[3] Amrouche, C.; Girault, V.; Giroire, J., Espaces de Sobolev avec poids et équation de Laplace dans \(R\)^{n} II, C. R. acad. sci. Paris, 315, 889-894, (1992), série I · Zbl 0793.46013
[4] Amrouche, C.; Girault, V.; Giroire, J., Weighted Sobolev spaces for Laplace’s equation in \(R\)^{n}, J. math. pures et appl., 73, 579-606, (1994) · Zbl 0836.35038
[5] Avantaggiati, A., Spazi di Sobolev con peso ed alcune applicazioni, Bolletino U.M.I., 13-A, 5, 1-52, (1976) · Zbl 0355.46016
[6] Babuška, I., The finite element method with Lagrangian multipliers, Numer. math., 20, 179-192, (1973) · Zbl 0258.65108
[7] Bartnik, R., The mass of an asymptotically flat manifold, Comm. on pure and applied math., Vol. XXXIX, 661-693, (1986) · Zbl 0598.53045
[8] Bolley, P.; Camus, J., Quelques résultats sur LES espaces de Sobolev avec poids, () · Zbl 0175.40004
[9] Bolley, P.; Pham, T.L., Propriétés d’indice en théorie höldérienne pour des opérateurs différentiels elliptiques dans \(R\)^{n}, J. math. pures et appl., 72, 105-119, (1993) · Zbl 0743.47009
[10] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O., anal. numér., R2, 129-151, (1974) · Zbl 0338.90047
[11] Cantor, M., Spaces of functions with asymptotic conditions on \(R\)^{n}, Indiana univ. math. J., 9, 24, 897-902, (1975) · Zbl 0441.46028
[12] Cantor, M., Boundary value problems for asymptotically homogeneous elliptic second order operators, J. of differential equations, 9, 34, 102-113, (1979) · Zbl 0388.35023
[13] Choquet-Bruhat, Y.; Christodoulou, D., Elliptic systems in Hs,δ spaces on manifolds which are Euclidean at infinity, Acta Mathematica, 129-150, (1981) · Zbl 0484.58028
[14] Dautray, R.; Lions, J.L., (), 553-701, Chapter XI
[15] Deny, J.; Lions, J.L., LES espaces du type de beppo Levi, Ann. inst. Fourier, Grenoble, 5, 305-370, (1954) · Zbl 0065.09903
[16] Farwig, R., A variational approach in weighted Sobolev spaces to the operator − δ + ∂/∂ξ1 in exterior domains of \(R\)3, Math. Z., 210, 449-464, (1992) · Zbl 0727.35041
[17] Fortunato, D., On the index of elliptic partial differential operators in \(R\)^{n}, Annali mat. pura appl., 119, 4, 317-331, (1979) · Zbl 0413.35029
[18] Giroire, J., Etude de quelques problèmes aux limites extérieurs et Résolution par equations intégrales, ()
[19] Giroire, J.; Nedelec, J.C., Numerical solution of an exterior Neumann problem using a double layer potential, Math. of comp., 32, 144, 973-990, (1978) · Zbl 0405.65060
[20] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman Boston · Zbl 0695.35060
[21] Hanouzet, B., Espaces de Sobolev avec poids - application au problème de Dirichlet dans un demi-espace, Rend. del sem. mat. Della univ. di Padova, XLVI, 227-272, (1971) · Zbl 0247.35041
[22] Hsiao, G.; Wendland, W., A finite element method for some integral equations of the first kind, J. of math. anal. and appl., 58, 3, 449-481, (1977) · Zbl 0352.45016
[23] Kudrjavcev, L.D., Direct and inverse imbedding theorems, (), 1-182 · Zbl 0095.09202
[24] Kufner, A., Weighted Sobolev spaces, (1985), Wiley Chichester · Zbl 0567.46009
[25] Kufner, A.; Opic, B., Hardy-type inequalities, (1990), Wiley New York · Zbl 0698.26007
[26] Leroux, M.N., Résolution numérique du problème du potentiel dans le plan par une Méthode variationnelle d’eléments finis, ()
[27] Leroux, M.N., Méthode d’éléments finis pour la résolution de problèmes extérieurs en dimension deux, R.a.i.r.o., 11, 27-60, (1977) · Zbl 0382.65055
[28] Lions, J.L.; Magenes, E., Problemi ai limiti non omogenei, V. ann. scuola norm. sup. Pisa, 16, 1-44, (1962) · Zbl 0115.31401
[29] Lizorkin, P.I., The behavior at infinity of functions in Liouville classes, (), 185-209 · Zbl 0476.46033
[30] Lockhart, R.B., Fredholm properties of a class of elliptic operators on non-compact manifolds, Duke math. J., 48, 1, 289-312, (1981) · Zbl 0486.35027
[31] Lockhart, R.B.; McOwen, R.C., On elliptic systems in \(R\)^{n}, Acta Mathematica, 150, 125-135, (1983) · Zbl 0517.35031
[32] McOwen, R.C., The behavior of the Laplacian on weighted Sobolev spaces, Comm. on pure and applied math., XXXII, 783-795, (1979) · Zbl 0426.35029
[33] McOwen, R.C., Boundary value problems for the Laplacian in an exterior domain, Comm. in partial differential equations, 7, 6, 783-798, (1981) · Zbl 0473.35036
[34] Murata, M., Isomorphism theorems for elliptic operators in \(R\)^{n}, Comm. in partial differential equations, 9, 11, 1085-1105, (1984) · Zbl 0556.47027
[35] Nečas, J., LES Méthodes directes en théorie des equations elliptiques, (1967), Masson Paris · Zbl 1225.35003
[36] Nedelec, J.C., Approximation par double couche du problème de Neumann extérieur, C. R. acad. sci. Paris, 286, 103-106, (1978), Série A · Zbl 0375.65047
[37] Nedelec, J.C.; Planchard, J., Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans \(R\)^{3}, R.A.I.R.O., anal. numér., R3, 105-129, (1973) · Zbl 0277.65074
[38] Nirenberg, L.; Walker, H.F., The null spaces of elliptic partial differential operators in \(R\)^{n}, Journal of mathematical analysis and applications, 42, 271-301, (1973) · Zbl 0272.35029
[39] Varnhorn, W., The Poisson equation with weights in exterior domains of \(R\)^{n}, Applicable analysis, 43, 135-145, (1992) · Zbl 0719.35020
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