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On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems


Author: Mirza Karamehmedović
Journal: Proc. Amer. Math. Soc. 143 (2015), 1515-1530
MSC (2010): Primary 35S05
DOI: https://doi.org/10.1090/S0002-9939-2014-12103-4
Published electronically: December 22, 2014
MathSciNet review: 3314066
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Abstract: We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary. To this end we identify and characterise a special subspace of the standard pseudodifferential operators with real-analytic symbols. The result is applicable in the estimation of the domain of analytic continuation of solutions across planar pieces of the boundary.


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  • [1] Louis Boutet de Monvel, Opérateurs pseudo-différentiels analytiques et opérateurs d’ordre infini, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 229–268 (French). MR 0341189
  • [2] A. Doicu, Yu. Eremin, and T. Wriedt, Acoustic & Electromagnetic Scattering Analysis Using Discrete Sources, Academic Press, 2000.
  • [3] Graeme Fairweather and Andreas Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998), no. 1-2, 69–95. Numerical treatment of boundary integral equations. MR 1662760, https://doi.org/10.1023/A:1018981221740
  • [4] Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
  • [5] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
    Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278
  • [6] Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
    Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. MR 781537
  • [7] R. Kangro and U. Kangro, Extendability of solutions of Helmholtz's equation across a hyperplane in three dimensions, Proc. of the International Conf. on Computational and Mathematics Methods in Science and Engineering, CMMSE 2006 (Madrid), 21-25 September 2006, pp. 433-437.
  • [8] M. Karamehmedović, P.-E. Hansen, K. Dirscherl, E. Karamehmedović, and T. Wriedt, Profile estimation for Pt submicron wire on rough Si substrate from experimental data, Opt. Express 20 (2012), 21678-21686.
  • [9] M. Karamehmedović, P.-E. Hansen, and T. Wriedt, An efficient scattering model for PEC and penetrable nanowires on a dielectric substrate, J. Eur. Opt. Soc. Rapid 6 (2011), 11021.
  • [10] -, A fast inversion method for highly conductive submicron wires on a substrate, J. Eur. Opt. Soc. Rapid 6 (2011), 11039.
  • [11] M. Karamehmedović, M.-P. Sørensen, P.-E. Hansen, and A. V. Lavrinenko, Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy, J. Eur. Opt. Soc. Rapid 5 (2010), 10021.
  • [12] Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623
  • [13] John M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, New York, 2003. MR 1930091
  • [14] R. F. Millar, The location of singularities of two-dimensional harmonic functions. I. Theory, SIAM J. Math. Anal. 1 (1970), 333–344. MR 0275759, https://doi.org/10.1137/0501030
  • [15] R. F. Millar, The location of singularities of two-dimensional harmonic functions. II. Applications, SIAM J. Math. Anal. 1 (1970), 345–353. MR 0275760, https://doi.org/10.1137/0501031
  • [16] R. F. Millar, Singularities of solutions to linear, second order analytic elliptic equations in two independent variables. I. The completely regular boundary., Applicable Anal. 1 (1971), no. 2, 101–121. MR 0287132, https://doi.org/10.1080/00036817108839009
  • [17] R. F. Millar, Singularities of two-dimensional exterior solutions of the Helmholtz equation, Proc. Cambridge Philos. Soc. 69 (1971), 175–188. MR 0268534
  • [18] R. F. Millar, Singularities of solutions to linear, second order, analytic elliptic equations in two independent variables. II. The piecewise regular boundary, Applicable Anal. 2 (1972/73), 301–320. MR 0390483, https://doi.org/10.1080/00036817208839046
  • [19] R. F. Millar, Singularities of solutions to exterior analytic boundary value problems for the Helmholz equation in three independent variables. I. The plane boundary, SIAM J. Math. Anal. 7 (1976), no. 1, 131–156. MR 0390462, https://doi.org/10.1137/0507012
  • [20] R. F. Millar, Singularities of solutions to exterior analytic boundary value problems for the Helmholtz equation in three independent variables. II. The axisymmetric boundary, SIAM J. Math. Anal. 10 (1979), no. 4, 682–694. MR 533939, https://doi.org/10.1137/0510063
  • [21] Robert F. Millar, The analytic continuation of solutions to elliptic boundary value problems in two independent variables, J. Math. Anal. Appl. 76 (1980), no. 2, 498–515. MR 587358, https://doi.org/10.1016/0022-247X(80)90045-1
  • [22] Robert F. Millar, Singularities and the Rayleigh hypothesis for solutions to the Helmholtz equation, IMA J. Appl. Math. 37 (1986), no. 2, 155–171. MR 983524, https://doi.org/10.1093/imamat/37.2.155
  • [23] Raul Kangro, Urve Kangro, and Roy Nicolaides, Extendability of solutions of Helmholtz’s equation to the interior of a two-dimensional scatterer, Quart. Appl. Math. 58 (2000), no. 3, 591–600. MR 1770657, https://doi.org/10.1090/qam/1770657
  • [24] Boris Sternin and Victor Shatalov, Differential equations on complex manifolds, Mathematics and its Applications, vol. 276, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1391964
  • [25] B. Yu. Sternin and V. E. Shatalov, Analytic continuation of solutions of integral equations and localization of singularities, Differ. Uravn. 32 (1996), no. 11, 1544–1553 (Russian, with Russian summary); English transl., Differential Equations 32 (1996), no. 11, 1541–1549 (1997). MR 1607109

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Additional Information

Mirza Karamehmedović
Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, DK-2800 Kgs. Lyngby, Denmark

DOI: https://doi.org/10.1090/S0002-9939-2014-12103-4
Received by editor(s): May 19, 2010
Received by editor(s) in revised form: November 17, 2012
Published electronically: December 22, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society