Boundary value problems and layer potentials on manifolds with cylindrical ends

Marius Mitrea; Victor Nistor

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 4, page 1151-1197
  • ISSN: 0011-4642

Abstract

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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.”

How to cite

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Mitrea, Marius, and Nistor, Victor. "Boundary value problems and layer potentials on manifolds with cylindrical ends." Czechoslovak Mathematical Journal 57.4 (2007): 1151-1197. <http://eudml.org/doc/31187>.

@article{Mitrea2007,
abstract = {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.”},
author = {Mitrea, Marius, Nistor, Victor},
journal = {Czechoslovak Mathematical Journal},
keywords = {layer potentials; manifolds with cylindrical ends; Dirichlet problem; layer potentials; manifolds with cylindrical ends; Dirichlet problem},
language = {eng},
number = {4},
pages = {1151-1197},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundary value problems and layer potentials on manifolds with cylindrical ends},
url = {http://eudml.org/doc/31187},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Mitrea, Marius
AU - Nistor, Victor
TI - Boundary value problems and layer potentials on manifolds with cylindrical ends
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 4
SP - 1151
EP - 1197
AB - 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.”
LA - eng
KW - layer potentials; manifolds with cylindrical ends; Dirichlet problem; layer potentials; manifolds with cylindrical ends; Dirichlet problem
UR - http://eudml.org/doc/31187
ER -

References

top
  1. Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Documenta Mathematica 11 (2006), 161–206 (electronic). (2006) MR2262931
  2. 10.1081/PDE-120037329, Comm. Partial Differential Equations 29 (2004), 671–705. (2004) MR2059145DOI10.1081/PDE-120037329
  3. 10.4007/annals.2007.165.717, Annals of Math. 165 (2007), 717–747. (2007) MR2335795DOI10.4007/annals.2007.165.717
  4. Collocation versus Galerkin procedures for boundary integral methods, In Boundary element methods in engineering, Springer, Berlin, 1982, pp. 18–33. (1982) MR0737197
  5. 10.1016/0045-7825(84)90151-8, Comput. Mthods Appl. Mech. Engrg. 45 (1984), 57–96. (1984) MR0759804DOI10.1016/0045-7825(84)90151-8
  6. 10.1007/s00211-005-0588-3, Numerische Mathematik 100 (2005), 165–184. (2005) MR2135780DOI10.1007/s00211-005-0588-3
  7. Differential operators on conic manifolds: Maximal regularity and parabolic equations, preprint, . MR1904055
  8. 10.2307/1971407, Annals of Math. 125 (1987), 437–465. (1987) MR0890159DOI10.2307/1971407
  9. 10.1016/0022-1236(92)90010-G, J. Funct. Anal. 109 (1992), 22–51. (1992) MR1183603DOI10.1016/0022-1236(92)90010-G
  10. 10.1512/iumj.1977.26.26007, Indiana Univ. Math. J. 26 (1977), 95–114. (1977) MR0432899DOI10.1512/iumj.1977.26.26007
  11. 10.1016/0022-1236(76)90076-8, J. Funct. Anal 21 (1976), 187–194. (1976) MR0394311DOI10.1016/0022-1236(76)90076-8
  12. 10.1007/BF02545747, Acta Math. 141 (1978), 165–186. (1978) MR0501367DOI10.1007/BF02545747
  13. 10.1353/ajm.2003.0012, Amer. J. Math. 125 (2003), 357–408. (2003) MR1963689DOI10.1353/ajm.2003.0012
  14. Functional calculus of pseudodifferential boundary value problems. Second edition, Progress in Mathematics 65, Birkhäuser, Boston, 1996. (1996) MR1385196
  15. The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941. (1941) Zbl0024.39703MR0003947
  16. Layer potentials on boundaries with corners and edges, Časopis Pěst. Mat. 113 (1988), 387–402. (1988) MR0981880
  17. 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. (1988) MR0973877
  18. Some examples concerning applicability of the Fredholm-Radon method in potential theory, Apl. Mat. 31 (1986), 293–308. (1986) MR0854323
  19. Integral operators in potential theory, Lecture Notes in Mathematics 823, Springer-Verlag, Berlin, 1980. (1980) MR0590244
  20. 10.2307/1969552, Annals of Math. 50 (1949), 587–665. (1949) Zbl0034.20502MR0031148DOI10.2307/1969552
  21. 10.2307/1970119, Annals of Math. 66 (1957), 89–140. (1957) MR0087879DOI10.2307/1970119
  22. Boundary value problems for elliptic equations in domains with conical or angular points, Transl. Moscow Math. Soc. 16 (1967), 227–313. (1967) MR0226187
  23. Spectral problems associated with corner singularities of solutions of elliptic equations, Mathematical Surveys and Monographs 85, AMS, Providence, RI, 2001. (2001) MR1788991
  24. Pseudodifferential analysis on continuous family groupoids, Documenta Math. (2000), 625–655 (electronic). (2000) MR1800315
  25. 10.1081/PDE-100001754, Commun. Partial Differ. Equations 26 (2001), 233–283. (2001) MR1842432DOI10.1081/PDE-100001754
  26. 10.1080/03605308308820276, Comm. Part. Diff. Eq. 8 (1983), 477–544. (1983) MR0695401DOI10.1080/03605308308820276
  27. Boundary problems for Dirac type operators on manifolds with multi-cylindrical end boundaries, Ann. Global Anal. Geom. 3 (2006), 337–383. (2006) MR2249562
  28. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A. K. Peters, Ltd., MA, 1993. (1993) Zbl0796.58050MR1348401
  29. Geometric scattering theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995. (1995) Zbl0849.58071MR1350074
  30. Elliptic operators of totally characteristic type, MSRI Preprint, 1983. (1983) 
  31. K -Theory of C * -algebras of b -pseudodifferential operators, Geom. Funct. Anal. 8 (1998), 99–122. (1998) MR1601850
  32. 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. (2004) MR2072504
  33. 10.1006/jfan.1998.3383, J. Funct. Anal. 163 (1999), 181–251. (1999) MR1680487DOI10.1006/jfan.1998.3383
  34. 10.1006/jfan.2000.3619, J. Funct. Anal. 176 (2000), 1–79. (2000) MR1781631DOI10.1006/jfan.2000.3619
  35. Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds, Memoirs of the Amer. Math. Soc. 150, 2001. (2001) MR1809655
  36. Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967. (1967) MR0227584
  37. 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. (2005) MR2114493
  38. Methods of modern mathematical physics, I, second edition, Academic Press Inc., New York, 1980. (1980) MR0751959
  39. Principles of functional analysis, Graduate Studies in Mathematics 36, second edition, AMS, Providence, RI, 2002. (2002) MR1861991
  40. 10.1007/BF00136867, Ann. Global Anal. and Geometry 10 (1992), 237–254. (1992) Zbl0788.47046MR1186013DOI10.1007/BF00136867
  41. Fréchet algebra technique for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance, Math. Nach. (1999), 145–185. (1999) MR1676318
  42. 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. (1995) MR1389012
  43. 10.1007/BF01202533, Integr. Equat. Oper. Theory 41 (2001), 93–114. (2001) MR1844462DOI10.1007/BF01202533
  44. Boundary value problems and singular pseudo-differential operators, John Wiley & Sons, Chichester-New York-Weinheim, 1998. (1998) Zbl0907.35146MR1631763
  45. Pseudodifferential operators and spectral theory, Springer Verlag, Berlin-Heidelberg-New York, 1987. (1987) Zbl0616.47040MR0883081
  46. Spectral theory of elliptic operators on non-compact manifolds, Astérisque 207 (1992), 35–108. (1992) Zbl0793.58039MR1205177
  47. Pseudodifferential operators, Princeton Mathematical Series 34, Princeton University Press, Princeton, N.J., 1981. (1981) Zbl0453.47026MR0618463
  48. Partial differential equations, Applied Mathematical Sciences, vol. II, Springer-Verlag, New York, 1996. (1996) MR1395147
  49. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs 81, Amer. Math. Soc., 2000. (2000) MR1766415
  50. 10.1016/0022-1236(84)90066-1, J. Funct. Anal. 59 (1984), 572–611. (1984) MR0769382DOI10.1016/0022-1236(84)90066-1

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