Mixed boundary value problems for elliptic equations of the second order in three-dimensional nonregular domains

Yu.M.Sybil

We consider mixed boundary value problems for elliptic equations of the second order in $\mathbb{R}^3$, i.e. interior, exterior and problems when boundary conditions are given on an open Lipschitz surface. We show existence and uniqueness of solutions of the posed problems in appropriate functional spaces. By means of integral representation these differential problems are reduced to systems of integral and integro-differential equations with the same type for all problems. The equivalence of obtained systems and given differential problems has been proved. We have also shown that matrix operators over the boundary of the domain in all three cases are positive definite and this gives us a possibility to construct a steady numerical algorithm using Galerkin method.

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3. Costabel M., Stephan E.P. An improved boundary element Galerkin method for three-dimensional crack problems. Integral Equation and Operator Theory. 10 (1987), 467--504.

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6. von Petersdorff T. Boundary integral equations for the mixed, Dirichlet, Neumann and transmission problems. Math. Meth. in Appl. Sci. 11 (1989), 185--213.

7. Schneider R. Reduction of order for pseudodifferential operators on Lipschitz domain. Commun. in Partial Differ. Equations. 16 (1991), 1263--1286.

8. Stephan E.P. Boundary integral equations for the screen problems in $R^3$. Int. Equat. Oper. Theory. 10 (1987), 236--257.

9. Stephan E.P. Boundary integral equations for the mixed boundary value problems in $R^3$. Math. Nachr. 134 (1987), 21--53.

10. Sybil Yu. Three-dimensional elliptic boundary value problems for an open Lipschitz surface. Matematychni Studii. 8, №1 (1997), 79--96.

11. Wendland W.L., Stephan E.P., G.C.Hsiao G.C. On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci. 1 (1979), 265--321.

12. Zaremba M.S. Sur un probl$\grave{e}$me mixte relatif a l'$\acute{e}$quation de Laplace. Bull. int. df l'Academie de Cracovie, Class. math. et nat. Ser. A. (1910), 313--344.

	Mixed boundary value problems for elliptic equations of the second order in three-dimensional nonregular domains
Author	Yu.M.Sybil Ivan Franko National University of L'viv ,Faculty of Applied Mathematics and Informatics
Abstract	We consider mixed boundary value problems for elliptic equations of the second order in $\mathbb{R}^3$, i.e. interior, exterior and problems when boundary conditions are given on an open Lipschitz surface. We show existence and uniqueness of solutions of the posed problems in appropriate functional spaces. By means of integral representation these differential problems are reduced to systems of integral and integro-differential equations with the same type for all problems. The equivalence of obtained systems and given differential problems has been proved. We have also shown that matrix operators over the boundary of the domain in all three cases are positive definite and this gives us a possibility to construct a steady numerical algorithm using Galerkin method.
Keywords	mixed boundary value problem, elliptic equation, nonregular domain
DOI	doi:10.30970/ms.28.2.191-205
Reference	1. Aubin J.-P. Approximation of elliptic boundary-value problems. Wiley-Interscience, New York, 1972. 2. Costabel M. Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19 (1988), 613--626. 3. Costabel M., Stephan E.P. An improved boundary element Galerkin method for three-dimensional crack problems. Integral Equation and Operator Theory. 10 (1987), 467--504. 4. Eskin G.I. Boundary problems for elliptic pseudo-differential operators. Transl. of Math. Mon., American Mathematical Society, Providence, Rhode Island. V. 52. Providence, Rhode Island, 1981. 5. Lions J.L., Magenes E. Nonhomogeneuos boundary-value problems and applications. V.1. Springer-Verlag, Berlin, 1972. 6. von Petersdorff T. Boundary integral equations for the mixed, Dirichlet, Neumann and transmission problems. Math. Meth. in Appl. Sci. 11 (1989), 185--213. 7. Schneider R. Reduction of order for pseudodifferential operators on Lipschitz domain. Commun. in Partial Differ. Equations. 16 (1991), 1263--1286. 8. Stephan E.P. Boundary integral equations for the screen problems in $R^3$. Int. Equat. Oper. Theory. 10 (1987), 236--257. 9. Stephan E.P. Boundary integral equations for the mixed boundary value problems in $R^3$. Math. Nachr. 134 (1987), 21--53. 10. Sybil Yu. Three-dimensional elliptic boundary value problems for an open Lipschitz surface. Matematychni Studii. 8, №1 (1997), 79--96. 11. Wendland W.L., Stephan E.P., G.C.Hsiao G.C. On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci. 1 (1979), 265--321. 12. Zaremba M.S. Sur un probl$\grave{e}$me mixte relatif a l'$\acute{e}$quation de Laplace. Bull. int. df l'Academie de Cracovie, Class. math. et nat. Ser. A. (1910), 313--344.
Pages	191-205
Volume	28
Issue	2
Year	2007
Journal	Matematychni Studii
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