Abstract
The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory.
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References
L. V. Ahlfors, Remarks on the Neumann-Poincaré integral equation, Pacific J. Math. 3 (1952), 271–280.
H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, American Mathematical Society, Providence, RI, 2009.
S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205–249.
J. Bremer, A fast direct solver for the integral equations of scattering theory on planar curves with corners, J. Comput. Phys. 231 (2012), 1879–1899.
T. Carleman, Über das Neumann-Poincarésche Problem für ein Gebiet mit Ecken, Almqvist and Wiksels, Uppsala, 1916.
T. Chang and K. Lee, Spectral properties of the layer potentials on Lipschitz domains, Illinois J. Math. 52 (2008), 463–472.
F. Cobos and T. Kühn, Eigenvalues of weakly singular integral operators, J. London Math. Soc. (2) 41 (1990), 323–335.
R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), 361–387.
J. Helsing and K-M. Perfekt, On the polarizability and capacitance of the cube, Appl. Comput. Harmon. Anal. 34 (2013), 445–468.
E. Johnston, A “counterexample” for the Schwarz-Christoffel transform, Amer. Math. Monthly 90 (1983), 701–703.
D. Khavinson, M. Putinar, and H. S. Shapiro, Poincaré’s variational problem in potential theory, Arch. Ration. Mech. Anal. 185 (2007), 143–184.
S. Krushkal, Fredholm eigenvalues of Jordan curves: geometric, variational and computational aspects, Analysis and Mathematical Physics, Birkhäuser, Basel, 2009, pp. 349–368.
R. Kühnau, Möglichst konforme Spiegelung an einer Jordankurve, Jahresber. Deutsch. Math.-Verein. 90 (1988), 90–109.
L. Lichtenstein, Neuere Entwicklung der Potentialtheorie; Konforme Abbildung, Encyklop. d. math. Wissensch. II C 3, Teubner, Leipzig, 1910.
S. E. Mikhailov, About traces, extensions, and co-normal derivative operators on Lipschitz domains, Integral Methods in Science and Engineering, Birkhäuser Boston, Boston, MA, 2008, pp. 149–160.
I. Mitrea, On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons, J. Fourier Anal. Appl. 8 (2002), 443–487.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
Ch. Pommerenke, Conformal maps at the boundary, Handbook of Complex Analysis: Geometric Function Theory, Vol. 1, North-Holland, Amsterdam, 2002, pp. 37–74.
G. Schober, Estimates for Fredholm eigenvalues based on quasiconformal mapping, Numerische, insbesondere approximationstheoretische Behandlung von Funktionalgleichungen, Lecture Notes in Math. 333, Springer, Berlin, 1973, pp. 211–217.
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572–611.
S. Werner, Spiegelungskoeffizient und Fredholmscher Eigenwert für gewisse Polygone, Ann. Acad. Sci. Fenn. Math. 22 (1997), 165–186.
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The work of the first author was partially supported by The Royal Physiographic Society in Lund.
The work of the second author was partially supported by the National Science Foundation Grant DMS-10-01071.
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Perfekt, KM., Putinar, M. Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. JAMA 124, 39–57 (2014). https://doi.org/10.1007/s11854-014-0026-5
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DOI: https://doi.org/10.1007/s11854-014-0026-5