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2013 | 11 | 1 | 27-54

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Layer potentials C*-algebras of domains with conical points

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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.


  • [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
  • [2] Alldridge A., Johansen, T.R., An index theorem for Wiener-Hopf operators, Adv. Math., 2008, 218(1), 163–201
  • [3] Amann H., Function spaces on singular manifolds, Math. Nachr. (in press), DOI: 10.1002/mana.201100157
  • [4] Ammann B., Ionescu A., Nistor V., Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math., 2006, 11, 161–206
  • [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
  • [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
  • [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
  • [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
  • [9] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994
  • [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] Debord C., Lescure J.-M., K-duality for pseudomanifolds with an isolated singularity, C. R. Math. Acad. Sci. Paris, 2003, 336(7), 577–580
  • [12] Debord C., Lescure J.-M., K-duality for pseudomanifolds with isolated singularities, J. Funct. Anal., 2005, 219(1), 109–133
  • [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
  • [14] Egorov Yu.V., Schulze B.-W., Pseudo-Differential Operators, Singularities, Applications, Oper. Theory Adv. Appl., 93, Birkhäuser, Basel, 1997
  • [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
  • [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
  • [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
  • [18] Folland G.B., Introduction to Partial Differential Equations, Princeton University Press, Princeton, 1995
  • [19] Karoubi M., Homologie cyclique et K-théorie, Astérisque, 1987, 149
  • [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] Kress R., Linear Integral Equations, Appl. Math. Sci., 82, Springer, New York, 1999
  • [22] Lauter R., Monthubert B., Nistor V., Pseudodifferential analysis on continuous family groupoids, Doc. Math., 2000, 5, 625–655
  • [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
  • [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
  • [25] Lescure J.-M., Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds, J. K-Theory, 2009, 4(2), 263–297
  • [26] Lewis J.E., Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains, Trans. Amer. Math. Soc., 1990, 320(1), 53–76
  • [27] Lewis J.E., Parenti C., Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations, 1983, 8(5), 477–544
  • [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
  • [29] Mackenzie K., Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lecture Note Ser., 124, Cambridge University Press, Cambridge, 1987
  • [30] Maz’ya V.G., Boundary integral equations, In: Analysis. IV, Encyclopaedia Math. Sci., 27, Springer, Berlin, 1991, 127–222
  • [31] Maz’ya V., Rossmann, J., Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr., 162, American Mathematical Society, Providence, 2010
  • [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
  • [33] Melrose R.B., The Atiyah-Patodi-Singer Index Theorem, Res. Notes in Math., 4, Peters, Wellesley, 1993
  • [34] Melrose R.B., Geometric Scattering Theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995
  • [35] Melrose R., Nistor V., K-theory of C*-algebras of b-pseudodifferential operators, Geom. Funct. Anal., 1998, 8(1), 88–122
  • [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
  • [37] Mitrea I., On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons, J. Fourier Anal. Appl., 2002, 8(5), 443–487
  • [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
  • [39] Moerdijk I., Mrčun J., Introduction to Foliations and Lie Groupoids, Cambridge Stud. Adv. Math., 91, Cambridge University Press, Cambridge, 2003
  • [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
  • [41] Monthubert B., Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal., 2003, 199(1), 243–286
  • [42] Monthubert B., Nistor V., A topological index theorem for manifolds with corners, Compos. Math., 2012, 148(2), 640–668
  • [43] Monthubert B., Pierrot F., Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math., 1997, 325(2), 193–198
  • [44] Moroianu S., K-theory of suspended pseudo-differential operators, K-Theory, 2003, 28(2), 167–181
  • [45] Muhly P.S., Renault J.N., C*-algebras of multivariate Wiener-Hopf operators, Trans. Amer. Math. Soc., 1982, 274(1), 1–44
  • [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] Nicola F., K-theory of SG-pseudo-differential algebras, Proc. Amer. Math. Soc., 2003, 131(9), 2841–2848
  • [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
  • [49] Nistor V., Weinstein A., Xu P., Pseudodifferential operators on differential groupoids, Pacific J. Math., 1999, 189(1), 117–152
  • [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
  • [51] Renault J.N., A Groupoid Approach to C*-Algebras, Lecture Notes in Math., 793, Springer, Berlin, 1980
  • [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
  • [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] 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
  • [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
  • [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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