Layer potentials C*-algebras of domains with conical points

Catarina Carvalho; Yu Qiao

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 27-54
  • ISSN: 2391-5455

Abstract

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To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.

How to cite

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Catarina Carvalho, and Yu Qiao. "Layer potentials C*-algebras of domains with conical points." Open Mathematics 11.1 (2013): 27-54. <http://eudml.org/doc/269473>.

@article{CatarinaCarvalho2013,
abstract = {To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.},
author = {Catarina Carvalho, Yu Qiao},
journal = {Open Mathematics},
keywords = {Layer potentials method; Conical domains; Desingularization; Groupoid C*-algebras; Weighted Sobolev spaces; Fredholmness; layer potentials method; conical domains; desingularization; groupoid -algebras; weighted Sobolev spaces},
language = {eng},
number = {1},
pages = {27-54},
title = {Layer potentials C*-algebras of domains with conical points},
url = {http://eudml.org/doc/269473},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Catarina Carvalho
AU - Yu Qiao
TI - Layer potentials C*-algebras of domains with conical points
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 27
EP - 54
AB - To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.
LA - eng
KW - Layer potentials method; Conical domains; Desingularization; Groupoid C*-algebras; Weighted Sobolev spaces; Fredholmness; layer potentials method; conical domains; desingularization; groupoid -algebras; weighted Sobolev spaces
UR - http://eudml.org/doc/269473
ER -

References

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  1. [1] Aastrup J., Melo S., Monthubert B., Schrohe E., Boutet de Monvel’s calculus and groupoids. I, J. Noncommut. Geom., 2010, 4(3), 313–329 http://dx.doi.org/10.4171/JNCG/57 Zbl1216.58008
  2. [2] Alldridge A., Johansen, T.R., An index theorem for Wiener-Hopf operators, Adv. Math., 2008, 218(1), 163–201 http://dx.doi.org/10.1016/j.aim.2007.11.024 Zbl1141.47044
  3. [3] Amann H., Function spaces on singular manifolds, Math. Nachr. (in press), DOI: 10.1002/mana.201100157 
  4. [4] Ammann B., Ionescu A., Nistor V., Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math., 2006, 11, 161–206 Zbl1247.35031
  5. [5] Ammann B., Lauter R., Nistor V., On the geometry of Riemannian manifolds with a Lie structure at infinity, Int. J. Math. Math. Sci., 2004, 4, 161–193 http://dx.doi.org/10.1155/S0161171204212108 Zbl1071.53020
  6. [6] Ammann B., Lauter R., Nistor V., Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math., 2007, 165(3), 717–747 http://dx.doi.org/10.4007/annals.2007.165.717 Zbl1133.58020
  7. [7] Băcuťă C., Mazzucato A., Nistor V., Zikatanov L., Interface and mixed boundary value problems on n-dimensional polyhedral domains, Doc. Math., 2010, 15, 687–745 Zbl1207.35117
  8. [8] Cannas da Silva A., Weinstein A., Geometric Models for Noncommutative Algebras, Berkeley Math. Lect. Notes, 10, American Mathematical Society, Providence, Berkeley Center for Pure and Applied Mathematics, Berkeley, 1999 Zbl1135.58300
  9. [9] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994 
  10. [10] Cordes H.O., On the technique of comparison algebra for elliptic boundary problems on noncompact manifolds, In: Operator Theory. 1, Durham, July 3–23, 1988, Proc. Sympos. Pure Math., 51(1), American Mathematical Society, Providence, 1990, 113–130 
  11. [11] Debord C., Lescure J.-M., K-duality for pseudomanifolds with an isolated singularity, C. R. Math. Acad. Sci. Paris, 2003, 336(7), 577–580 http://dx.doi.org/10.1016/S1631-073X(03)00124-9 Zbl1040.19001
  12. [12] Debord C., Lescure J.-M., K-duality for pseudomanifolds with isolated singularities, J. Funct. Anal., 2005, 219(1), 109–133 http://dx.doi.org/10.1016/j.jfa.2004.03.017 Zbl1065.58014
  13. [13] Debord C., Lescure J.-M., Nistor V., Groupoids and an index theorem for conical pseudo-manifolds, J. Reine Angew. Math., 2009, 628, 1–35 http://dx.doi.org/10.1515/CRELLE.2009.017 Zbl1169.58005
  14. [14] Egorov Yu.V., Schulze B.-W., Pseudo-Differential Operators, Singularities, Applications, Oper. Theory Adv. Appl., 93, Birkhäuser, Basel, 1997 http://dx.doi.org/10.1007/978-3-0348-8900-1 
  15. [15] Elschner J., The double layer potential operator over polyhedral domains. I. Solvability in weighted Sobolev spaces, Appl. Anal., 1992, 45(1–4), 117–134 http://dx.doi.org/10.1080/00036819208840092 Zbl0749.31002
  16. [16] Fabes E.B., Jodeit M. Jr., Lewis J.E., Double layer potentials for domains with corners and edges, Indiana Univ. Math. J., 1977, 26(1), 95–114 http://dx.doi.org/10.1512/iumj.1977.26.26007 Zbl0363.35010
  17. [17] Fabes E.B., Jodeit M. Jr., Rivière N.M., Potential techniques for boundary value problems on C 1-domains, Acta Math., 1978, 141(3–4), 165–186 http://dx.doi.org/10.1007/BF02545747 Zbl0402.31009
  18. [18] Folland G.B., Introduction to Partial Differential Equations, Princeton University Press, Princeton, 1995 Zbl0841.35001
  19. [19] Karoubi M., Homologie cyclique et K-théorie, Astérisque, 1987, 149 
  20. [20] Kondrat’ev V.A., Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 1967, 16, 209–292 (in Russian) 
  21. [21] Kress R., Linear Integral Equations, Appl. Math. Sci., 82, Springer, New York, 1999 http://dx.doi.org/10.1007/978-1-4612-0559-3 
  22. [22] Lauter R., Monthubert B., Nistor V., Pseudodifferential analysis on continuous family groupoids, Doc. Math., 2000, 5, 625–655 Zbl0961.22005
  23. [23] Lauter R., Nistor V., Analysis of geometric operators on open manifolds: a groupoid approach, In: Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001, 181–229 http://dx.doi.org/10.1007/978-3-0348-8364-1_8 
  24. [24] Le Gall P.-Y., Monthubert B., K-theory of the indicial algebra of a manifold with corners, K-Theory, 2001, 23(2), 105–113 http://dx.doi.org/10.1023/A:1017550814634 
  25. [25] Lescure J.-M., Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds, J. K-Theory, 2009, 4(2), 263–297 http://dx.doi.org/10.1017/is008007020jkt062 Zbl1183.58005
  26. [26] Lewis J.E., Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains, Trans. Amer. Math. Soc., 1990, 320(1), 53–76 http://dx.doi.org/10.1090/S0002-9947-1990-1005935-5 Zbl0711.35041
  27. [27] Lewis J.E., Parenti C., Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations, 1983, 8(5), 477–544 http://dx.doi.org/10.1080/03605308308820276 Zbl0532.35085
  28. [28] Li H., Mazzucato A., Nistor V., Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains, Electron. Trans. Numer. Anal., 2010, 37, 41–69 Zbl1205.65317
  29. [29] Mackenzie K., Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lecture Note Ser., 124, Cambridge University Press, Cambridge, 1987 http://dx.doi.org/10.1017/CBO9780511661839 Zbl0683.53029
  30. [30] Maz’ya V.G., Boundary integral equations, In: Analysis. IV, Encyclopaedia Math. Sci., 27, Springer, Berlin, 1991, 127–222 http://dx.doi.org/10.1007/978-3-642-58175-5_2 
  31. [31] Maz’ya V., Rossmann, J., Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr., 162, American Mathematical Society, Providence, 2010 
  32. [32] Mazzucato A., Nistor V., Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks, Arch. Ration. Mech. Anal., 2010, 195(1), 25–73 http://dx.doi.org/10.1007/s00205-008-0180-y Zbl1188.35189
  33. [33] Melrose R.B., The Atiyah-Patodi-Singer Index Theorem, Res. Notes in Math., 4, Peters, Wellesley, 1993 Zbl0796.58050
  34. [34] Melrose R.B., Geometric Scattering Theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995 
  35. [35] Melrose R., Nistor V., K-theory of C*-algebras of b-pseudodifferential operators, Geom. Funct. Anal., 1998, 8(1), 88–122 http://dx.doi.org/10.1007/s000390050049 Zbl0898.46060
  36. [36] Mitrea D., Mitrea I., On the Besov regularity of conformal maps and layer potentials on nonsmooth domains, J. Funct. Anal., 2003, 201(2), 380–429 http://dx.doi.org/10.1016/S0022-1236(03)00086-7 Zbl1060.45001
  37. [37] Mitrea I., On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons, J. Fourier Anal. Appl., 2002, 8(5), 443–487 http://dx.doi.org/10.1007/s00041-002-0022-5 Zbl1161.45304
  38. [38] Mitrea M., Nistor V., Boundary value problems and layer potentials on manifolds with cylindrical ends, Czechoslovak Math. J., 2007, 57(132)(4), 1151–1197 http://dx.doi.org/10.1007/s10587-007-0118-9 Zbl1174.31002
  39. [39] Moerdijk I., Mrčun J., Introduction to Foliations and Lie Groupoids, Cambridge Stud. Adv. Math., 91, Cambridge University Press, Cambridge, 2003 http://dx.doi.org/10.1017/CBO9780511615450 Zbl1029.58012
  40. [40] Monthubert B., Groupoids of manifolds with corners and index theory, In: Groupoids in Analysis, Geometry, and Physics, Boulder, June 20–24, 1999, Contemp. Math., 282, American Mathematical Society, Providence, 2001, 147–157 http://dx.doi.org/10.1090/conm/282/04684 
  41. [41] Monthubert B., Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal., 2003, 199(1), 243–286 http://dx.doi.org/10.1016/S0022-1236(02)00038-1 Zbl1025.58009
  42. [42] Monthubert B., Nistor V., A topological index theorem for manifolds with corners, Compos. Math., 2012, 148(2), 640–668 http://dx.doi.org/10.1112/S0010437X11005458 Zbl1247.58016
  43. [43] Monthubert B., Pierrot F., Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math., 1997, 325(2), 193–198 http://dx.doi.org/10.1016/S0764-4442(97)84598-3 
  44. [44] Moroianu S., K-theory of suspended pseudo-differential operators, K-Theory, 2003, 28(2), 167–181 http://dx.doi.org/10.1023/A:1024537820594 
  45. [45] Muhly P.S., Renault J.N., C*-algebras of multivariate Wiener-Hopf operators, Trans. Amer. Math. Soc., 1982, 274(1), 1–44 Zbl0509.46049
  46. [46] Muhly P.S., Renault J.N., Williams D.P., Equivalence and isomorphism for groupoid C*-algebras, J. Operator Theory, 1987, 17(1), 3–22 
  47. [47] Nicola F., K-theory of SG-pseudo-differential algebras, Proc. Amer. Math. Soc., 2003, 131(9), 2841–2848 http://dx.doi.org/10.1090/S0002-9939-03-06837-0 Zbl1031.46079
  48. [48] Nistor V., Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains, In: Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, Contemp. Math., 366, American Mathematical Society, Providence, 2005, 307–328 http://dx.doi.org/10.1090/conm/366/06734 
  49. [49] Nistor V., Weinstein A., Xu P., Pseudodifferential operators on differential groupoids, Pacific J. Math., 1999, 189(1), 117–152 http://dx.doi.org/10.2140/pjm.1999.189.117 Zbl0940.58014
  50. [50] Qiao Y., Nistor V., Single and double layer potentials on domains with conical points I: straight cones, Integral Equations and Operator Theory, 2012, 72(3), 419–448 http://dx.doi.org/10.1007/s00020-012-1947-y Zbl1267.47077
  51. [51] Renault J.N., A Groupoid Approach to C*-Algebras, Lecture Notes in Math., 793, Springer, Berlin, 1980 Zbl0433.46049
  52. [52] Rørdam M., Larsen F., Laustsen N., An Introduction to K-Theory for C*-Algebras, London Math. Soc. Stud. Texts, 49, Cambridge University Press, Cambridge, 2000 Zbl0967.19001
  53. [53] Schrohe E., Spaces of weighted symbols and weighted Sobolev spaces on manifolds, In: Pseudodifferential Operators, Oberwolfach, February 2–8, 1986, Lecture Notes in Math., 1256, Springer, Berlin, 1987, 360–377 
  54. [54] Schrohe E., Schulze B.-W., Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. I, In: Pseudo-Differential Calculus and Mathematical Physics, Math. Top., 5, Akademie, Berlin, 1994, 97–209 Zbl0827.35145
  55. [55] Schrohe E., Schulze B.-W., Boundary value problems in Boutet de Monvel’s algebra for manifolds with conical singularities. II, In: Boundary Value Problems, Schrödinger Operators, Deformation Quantization, Math. Top., 8, Akademie, Berlin, 1995, 70–205 Zbl0847.35156
  56. [56] Schulze B.-W., Boundary Value Problems and Singular Pseudo-Differential Operators, Pure Appl. Math. (N.Y.), John Wiley & Sons, Chichester, 1998 

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