| Title:
|
Boundary value problems and layer potentials on manifolds with cylindrical ends (English) |
| Author:
|
Mitrea, Marius |
| Author:
|
Nistor, Victor |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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57 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
1151-1197 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.” (English) |
| Keyword:
|
layer potentials |
| Keyword:
|
manifolds with cylindrical ends |
| Keyword:
|
Dirichlet problem |
| MSC:
|
31C12 |
| MSC:
|
35J05 |
| MSC:
|
35S15 |
| MSC:
|
47G30 |
| MSC:
|
58J05 |
| MSC:
|
58J32 |
| MSC:
|
58J40 |
| idZBL:
|
Zbl 1174.31002 |
| idMR:
|
MR2357585 |
| . |
| Date available:
|
2009-09-24T11:52:15Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128232 |
| . |
| Reference:
|
[1] B. Ammann, A. Ionescu and V. Nistor: Sobolev spaces on Lie manifolds and regularity for polyhedral domains.Documenta Mathematica 11 (2006), 161–206 (electronic). MR 2262931 |
| Reference:
|
[2] B. Ammann, R. Lauter, V. Nistor and A. Vasy: Complex powers and non-compact manifolds.Comm. Partial Differential Equations 29 (2004), 671–705. MR 2059145, 10.1081/PDE-120037329 |
| Reference:
|
[3] B. Ammann, R. Lauter and V. Nistor: Pseudodifferential operators on manifolds with a Lie structure at infinity.Annals of Math. 165 (2007), 717–747. MR 2335795, 10.4007/annals.2007.165.717 |
| Reference:
|
[4] D. Arnold and W. Wendland: Collocation versus Galerkin procedures for boundary integral methods, In Boundary element methods in engineering.Springer, Berlin, 1982, pp. 18–33. MR 0737197 |
| Reference:
|
[5] D. Arnold, I. Babuška and J. Osborn: Finite element methods: principles for their selection.Comput. Mthods Appl. Mech. Engrg. 45 (1984), 57–96. MR 0759804, 10.1016/0045-7825(84)90151-8 |
| Reference:
|
[6] C. Bacuta, V. Nistor and L. Zikatanov: Improving the convergence of ‘high order finite elements’ on polygons and domains with cusps.Numerische Mathematik 100 (2005), 165–184. MR 2135780, 10.1007/s00211-005-0588-3 |
| Reference:
|
[7] S. Coriasco, E. Schrohe and J. Seiler: Differential operators on conic manifolds: Maximal regularity and parabolic equations, preprint.. MR 1904055 |
| Reference:
|
[8] B. Dahlberg and C. Kenig: Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains.Annals of Math. 125 (1987), 437–465. MR 0890159, 10.2307/1971407 |
| Reference:
|
[9] A. Erkip and E. Schrohe: Normal solvability of elliptic boundary value problems on asymptotically flat manifolds.J. Funct. Anal. 109 (1992), 22–51. MR 1183603, 10.1016/0022-1236(92)90010-G |
| Reference:
|
[10] E.B. Fabes, M. Jodeit and J.E. Lewis: Double layer potentials for domains with corners and edges.Indiana Univ. Math. J. 26 (1977), 95–114. MR 0432899, 10.1512/iumj.1977.26.26007 |
| Reference:
|
[11] E.B. Fabes, M. Jodeit and J.E. Lewis: On the spectra of a Hardy kernel.J. Funct. Anal 21 (1976), 187–194. MR 0394311, 10.1016/0022-1236(76)90076-8 |
| Reference:
|
[12] E. Fabes, M. Jodeit and N. Riviere: Potential techniques for boundary value problems on $C^1$ domains.Acta Math. 141 (1978), 165–186. MR 0501367, 10.1007/BF02545747 |
| Reference:
|
[13] J. Gill and G. Mendoza: Adjoints of elliptic cone operators.Amer. J. Math. 125 (2003), 357–408. MR 1963689, 10.1353/ajm.2003.0012 |
| Reference:
|
[14] G. Grubb: Functional calculus of pseudodifferential boundary value problems. Second edition.Progress in Mathematics 65, Birkhäuser, Boston, 1996. MR 1385196 |
| Reference:
|
[15] W. V. D. Hodge: The Theory and Applications of Harmonic Integrals.Cambridge University Press, 1941. Zbl 0024.39703, MR 0003947 |
| Reference:
|
[16] T. Angell, R. Kleinman and J. Král: Layer potentials on boundaries with corners and edges.Časopis Pěst. Mat. 113 (1988), 387–402. MR 0981880 |
| Reference:
|
[17] J. Král: Potential theory-surveys and problems.Lecture Notes in Mathematics, 1344, Proceedings of the Conference on Potential Theory held in Prague, Král, J. and Lukeš, J. and Netuka, I. and Veselý, J. Eds., Springer, Berlin, 1988. MR 0973877 |
| Reference:
|
[18] J. Král and W. Wendland: Some examples concerning applicability of the Fredholm-Radon method in potential theory.Apl. Mat. 31 (1986), 293–308. MR 0854323 |
| Reference:
|
[19] J. Král: Integral operators in potential theory.Lecture Notes in Mathematics 823, Springer-Verlag, Berlin, 1980. MR 0590244 |
| Reference:
|
[20] K. Kodaira: Harmonic fields in Riemannian manifolds (Generalized potential theory).Annals of Math. 50 (1949), 587–665. Zbl 0034.20502, MR 0031148, 10.2307/1969552 |
| Reference:
|
[21] J. J. Kohn and D. C. Spencer: Complex Neumann problems.Annals of Math. 66 (1957), 89–140. MR 0087879, 10.2307/1970119 |
| Reference:
|
[22] V. A. Kondratiev: Boundary value problems for elliptic equations in domains with conical or angular points.Transl. Moscow Math. Soc. 16 (1967), 227–313. MR 0226187 |
| Reference:
|
[23] V. A. Kozlov, V. G. Mazya and J. Rossmann: Spectral problems associated with corner singularities of solutions of elliptic equations.Mathematical Surveys and Monographs 85, AMS, Providence, RI, 2001. MR 1788991 |
| Reference:
|
[24] R. Lauter, B. Monthubert and V. Nistor: Pseudodifferential analysis on continuous family groupoids.Documenta Math. (2000), 625–655 (electronic). MR 1800315 |
| Reference:
|
[25] R. Lauter and S. Moroianu: Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries.Commun. Partial Differ. Equations 26 (2001), 233–283. MR 1842432, 10.1081/PDE-100001754 |
| Reference:
|
[26] J. Lewis and C. Parenti: Pseudodifferential operators of Mellin type.Comm. Part. Diff. Eq. 8 (1983), 477–544. MR 0695401, 10.1080/03605308308820276 |
| Reference:
|
[27] P. Loya and J. Park: Boundary problems for Dirac type operators on manifolds with multi-cylindrical end boundaries.Ann. Global Anal. Geom. 3 (2006), 337–383. MR 2249562 |
| Reference:
|
[28] R. B. Melrose: The Atiyah-Patodi-Singer index theorem.Research Notes in Mathematics 4, A. K. Peters, Ltd., MA, 1993. Zbl 0796.58050, MR 1348401 |
| Reference:
|
[29] R. B. Melrose: Geometric scattering theory.Stanford Lectures, Cambridge University Press, Cambridge, 1995. Zbl 0849.58071, MR 1350074 |
| Reference:
|
[30] R. B. Melrose and G. Mendoza: Elliptic operators of totally characteristic type.MSRI Preprint, 1983. |
| Reference:
|
[31] R.B. Melrose and V. Nistor: $K$-Theory of $C^*$-algebras of $b$-pseudodifferential operators.Geom. Funct. Anal. 8 (1998), 99–122. MR 1601850 |
| Reference:
|
[32] M. Mitrea and V. Nistor: A note on boundary value problems on manifolds with cylindrical ends.In Aspects of boundary problems in analysis and geometry, Birkhäuser, Basel, 2004, pp. 472–494. MR 2072504 |
| Reference:
|
[33] M. Mitrea and M.E. Taylor: Boundary layer methods for Lipschitz domains in Riemannian manifolds.J. Funct. Anal. 163 (1999), 181–251. MR 1680487, 10.1006/jfan.1998.3383 |
| Reference:
|
[34] M. Mitrea and M. E. Taylor: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem.J. Funct. Anal. 176 (2000), 1–79. MR 1781631, 10.1006/jfan.2000.3619 |
| Reference:
|
[35] D. Mitrea, M. Mitrea and M. Taylor: Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds.Memoirs of the Amer. Math. Soc. 150, 2001. MR 1809655 |
| Reference:
|
[36] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Masson et Cie, Paris, 1967. MR 0227584 |
| Reference:
|
[37] V. Nistor: Pseudodifferential opearators on non-compact manifolds and analysis on polyhedral domains.Proceedings of the Workshop on Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, Roskilde University, Contemporary Mathematics, AMS, Rhode Island, 2005, pp. 307–328. MR 2114493 |
| Reference:
|
[38] M. Reed and B. Simon: Methods of modern mathematical physics.I, second edition, Academic Press Inc., New York, 1980. MR 0751959 |
| Reference:
|
[39] M. Schechter: Principles of functional analysis.Graduate Studies in Mathematics 36, second edition, AMS, Providence, RI, 2002. MR 1861991 |
| Reference:
|
[40] E. Schrohe: Spectral invariance, ellipticity and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces.Ann. Global Anal. and Geometry 10 (1992), 237–254. Zbl 0788.47046, MR 1186013, 10.1007/BF00136867 |
| Reference:
|
[41] E. Schrohe: Fréchet algebra technique for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance.Math. Nach. (1999), 145–185. MR 1676318 |
| Reference:
|
[42] E. Schrohe and B.-W. Schulze: Boundary Value Problems in Boutet de Monvel’s Algebra for Manifolds with Conical Singularities II.Boundary value problems, Schrödinger operators, deformation quantization, Math. Top. 8, Akademie Verlag, Berlin, 1995, pp. 70–205. MR 1389012 |
| Reference:
|
[43] E. Schrohe and J. Seiler: Ellipticity and invertibility in the cone algebra on $L_p$-Sobolev spaces.Integr. Equat. Oper. Theory 41 (2001), 93–114. MR 1844462, 10.1007/BF01202533 |
| Reference:
|
[44] B.-W. Schulze: Boundary value problems and singular pseudo-differential operators.John Wiley & Sons, Chichester-New York-Weinheim, 1998. Zbl 0907.35146, MR 1631763 |
| Reference:
|
[45] M. A. Shubin: Pseudodifferential operators and spectral theory.Springer Verlag, Berlin-Heidelberg-New York, 1987. Zbl 0616.47040, MR 0883081 |
| Reference:
|
[46] M. A. Shubin: Spectral theory of elliptic operators on non-compact manifolds.Astérisque 207 (1992), 35–108. Zbl 0793.58039, MR 1205177 |
| Reference:
|
[47] M. E. Taylor: Pseudodifferential operators.Princeton Mathematical Series 34, Princeton University Press, Princeton, N.J., 1981. Zbl 0453.47026, MR 0618463 |
| Reference:
|
[48] M. E. Taylor: Partial differential equations, Applied Mathematical Sciences, vol. II.Springer-Verlag, New York, 1996. MR 1395147 |
| Reference:
|
[49] M. E. Taylor: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs 81.Amer. Math. Soc., 2000. MR 1766415 |
| Reference:
|
[50] G. Verchota: Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains.J. Funct. Anal. 59 (1984), 572–611. MR 0769382, 10.1016/0022-1236(84)90066-1 |
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