Three-dimensional elliptic boundary value problems for an open
Lipschitz surface

Yu. Sybil

We consider Dirichlet and Neumann boundary value problems for elliptic equation of the second order in $\Bbb R^3$ when the boundary conditions are given on the open Lipschitz surface. Using the structure of certain Hilbert spaces and properties of corresponding trace maps we get the existence and uniqueness of the solution of initial boundary value problems and the solutions of obtained boundary equations over an open surface $S$ as well.

1. J. P. Aubin, Approximation of elliptic boundary-value problems, Wiley-Interscience, New York, 1972.

2. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), 613–626.

3. M. Costabel, Boundary integral operators for the heat equation, Integral Equation and Operator Theory 13 (1990), 498–552.

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

5. G.I. Eskin, Boundary problems for elliptic pseudo-differential operators, vol. 52, Transl. of Math. Mon., American Mathematical Society, Providence, Rhode Island, 1981.

6. Hayashi Y., The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary, J. Math. Anal. Appl. 44 (1973), 489–530.

7. J.L. Lions, E. Magenes, Nonhomogeneous boundary-value problems and applications, vol. 1, Springer-Verlag, Berlin, 1972.

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

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

10. E.P. Stephan, Boundary integral equations for the mixed boundary value problems in ℝ³, Math. Nachr. 134 (1987), 21–53.

11. E.P. Stephan, Boundary integral equations for the screen problems in ℝ³, Int. Equat. Oper. Theory 10 (1987), 236–257.

12. Sybil Yu., Existence of solutions of the integral equations of Dirichlet and Neumann problems in the case of open boundaries, Visn. of Lviv University. Ser. mech.-math. 29 (1988), 46–48. (Russian)

13. Sybil Yu., Bandrovskij S., The Dirichlet problem for the three-dimensional Helmholtz equation for an open boundaries, Theoreticheskaya Electrotekhnika 45 (1988), 22–27. (Russian)

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

	Three-dimensional elliptic boundary value problems for an open Lipschitz surface
Author	Yu. Sybil Lviv University
Abstract	We consider Dirichlet and Neumann boundary value problems for elliptic equation of the second order in $\Bbb R^3$ when the boundary conditions are given on the open Lipschitz surface. Using the structure of certain Hilbert spaces and properties of corresponding trace maps we get the existence and uniqueness of the solution of initial boundary value problems and the solutions of obtained boundary equations over an open surface $S$ as well.
Keywords	Dirichlet boundary value problem, Neumann boundary value problem, second-order elliptic equation, open Lipschitz surface, Hilbert spaces, trace maps, existence, uniqueness, boundary equations
DOI	doi:10.30970/ms.8.1.79-96
Reference	1. J. P. Aubin, Approximation of elliptic boundary-value problems, Wiley-Interscience, New York, 1972. 2. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), 613–626. 3. M. Costabel, Boundary integral operators for the heat equation, Integral Equation and Operator Theory 13 (1990), 498–552. 4. M. Costabel, E.P. Stephan, An improved boundary element Galerkin method for three-dimensional crack problems, Integral Equation and Operator Theory 10 (1987), 467–504. 5. G.I. Eskin, Boundary problems for elliptic pseudo-differential operators, vol. 52, Transl. of Math. Mon., American Mathematical Society, Providence, Rhode Island, 1981. 6. Hayashi Y., The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary, J. Math. Anal. Appl. 44 (1973), 489–530. 7. J.L. Lions, E. Magenes, Nonhomogeneous boundary-value problems and applications, vol. 1, Springer-Verlag, Berlin, 1972. 8. T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems, Math. Meth. in the Appl. Sci. 11 (1989), 185–213. 9. R. Schneider, Reduction of order for pseudodifferential operators on Lipschitz domain, Commun. in Partial Differ. Equations 16 (1991), 1263–1286. 10. E.P. Stephan, Boundary integral equations for the mixed boundary value problems in ℝ³, Math. Nachr. 134 (1987), 21–53. 11. E.P. Stephan, Boundary integral equations for the screen problems in ℝ³, Int. Equat. Oper. Theory 10 (1987), 236–257. 12. Sybil Yu., Existence of solutions of the integral equations of Dirichlet and Neumann problems in the case of open boundaries, Visn. of Lviv University. Ser. mech.-math. 29 (1988), 46–48. (Russian) 13. Sybil Yu., Bandrovskij S., The Dirichlet problem for the three-dimensional Helmholtz equation for an open boundaries, Theoreticheskaya Electrotekhnika 45 (1988), 22–27. (Russian) 14. W.L. Wendland, E.P. Stephan, G.C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci. 1, 265–321.
Pages	79-96
Volume	8
Issue	1
Year	1997
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Three-dimensional elliptic boundary value problems for an open Lipschitz surface