Advertisement

Communications in Mathematical Physics

, Volume 368, Issue 2, pp 799–842 | Cite as

The Limiting Absorption Principle for Periodic Differential Operators and Applications to Nonlinear Helmholtz Equations

  • Rainer MandelEmail author
Article

Abstract

We prove an Lp-version of the limiting absorption principle for a class of periodic elliptic differential operators of second order. The result is applied to the construction of nontrivial solutions of nonlinear Helmholtz equations with periodic coefficient functions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The author thanks the reviewer for several helpful remarks and corrections. Furthermore, he is grateful to TomášDohnal (Martin-Luther-UniversitätHalle-Wittenberg) for stimulating discussions about Floquet-Bloch theory during the past years and for providing Fig. 1. Additionally, he gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 ”Wave phenomena: analysis and numerics”.

References

  1. 1.
    Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Berezin, F., Shubin, M.: The Schrödinger equation, volume 66 of Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991)Google Scholar
  3. 3.
    Bloch F.: Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52(7), 550–600 (1929)ADSGoogle Scholar
  4. 4.
    Brillouin, L.: Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices. 2nd edn. Dover Publications, Inc., New York (1953)Google Scholar
  5. 5.
    Cacciafesta F., D’Ancona P., Lucà R.: Helmholtz and dispersive equations with variable coefficients on exterior domains. SIAM J. Math. Anal. 48(3), 1798–1832 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dohnal T., Uecker H: Coupled mode equations and gap solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential. Phys. D 238(9-10), 860–879 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eastham, M.: The Spectral Theory of Periodic Differential Equations. Texts in Mathematics (Edinburgh). Scottish Academic Press, Edinburgh; Hafner Press, New York (1973)Google Scholar
  8. 8.
    Èĭdus D.: On the principle of limiting absorption. Mat. Sb. (N.S.) 57((99), 13–44 (1962)MathSciNetGoogle Scholar
  9. 9.
    Evéquoz G.: Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane. Analysis 37(2), 55–68 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Evéquoz G., Weth T.: Dual variational methods and nonvanishing for the nonlinear Helmholtz equation. Adv. Math. 280, 690–728 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Floquet G.: Sur les équations différentielles linéaires à coefficients périodiques. Ann. Sci. École Norm. Sup. (2) 12, 47–88 (1883)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldberg M., Schlag W.: A limiting absorption principle for the three-dimensional Schrödinger equation with L p potentials. Int. Math. Res. Not. 75, 4049–4071 (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Goldman R.: Curvature formulas for implicit curves and surfaces. Comput. Aided Geom. Des. 22(7), 632–658 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gutiérrez S.: Non trivial L q solutions to the Ginzburg-Landau equation. Math. Ann. 328(1-2), 1–25 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Il’in, V., Joó, I.: Uniform estimation of eigenfunctions and an upper bound on the number of eigenvalues of the Sturm-Liouville operator with a potential from the class L p. Differentsial’nye Uravneniya, 15(7):1164–1174, 1340 (1979)Google Scholar
  16. 16.
    Kachkovskiĭ I.: The Stein-Tomas theorem for a torus and the periodic Schrödinger operator with singular potential. Algebra i Analiz 24(6), 124–138 (2012)Google Scholar
  17. 17.
    Kato, T.: A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York (1982)Google Scholar
  18. 18.
    Kenig C., Ruiz A., Sogge C.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Komornik V.: Uniformly bounded Riesz bases and equiconvergence theorems. Bol. Soc. Parana. Mat. (3) 25(1-2), 139–146 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kuchment, P.: Floquet Theory for Partial Differential Equations, volume 60 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1993)Google Scholar
  21. 21.
    Kuchment, P.: The mathematics of photonic crystals. In: Mathematical modeling in optical science, volume 22 of Frontiers Applied Mathematics (pp. 207–272). SIAM, Philadelphia (2001)Google Scholar
  22. 22.
    Lee, J.M.: Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. 2nd edn, Springer, New York (2013)Google Scholar
  23. 23.
    Littman W.: Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Am. Math. Soc. 69, 766–770 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups, volume 283 of Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton (2007)Google Scholar
  25. 25.
    Mandel, R.: A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems. Manuscr. Math. (2017)Google Scholar
  26. 26.
    Mandel, R., Montefusco, E., Pellacci, B.: Oscillating solutions for nonlinear helmholtz equations. Z. Angew. Math. Phys. 68(6):121, 10 (2017)Google Scholar
  27. 27.
    Odeh F.: Principles of limiting absorption and limiting amplitude in scattering theory. I. Schrödinger’s equation. J. Math. Phys. 2, 794–800 (1961)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Odeh F., Keller J.: Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys. 5, 1499–1504 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Perthame B., Vega L.: Morrey–Campanato estimates for Helmholtz equations. J. Funct. Anal. 164(2), 340–355 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Radosz, M.: The Principles of Limit Absorption and Limit Amplitude for Periodic Operators. Ph.D. thesis, KIT (2010)Google Scholar
  31. 31.
    Radosz M.: New limiting absorption and limit amplitude principles for periodic operators. Z. Angew. Math. Phys. 66(2), 253–275 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)Google Scholar
  33. 33.
    Rodnianski I., Tao T.: Effective limiting absorption principles, and applications. Commun. Math. Phys. 333(1), 1–95 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ruiz, A.: Harmonic analysis and inverse problems, lecture notes, (2013)Google Scholar
  35. 35.
    Sólyom, J.: Fundamentals of the Physics of Solids. Volume II: Electronic Properties. Springer, Berlin (2009)Google Scholar
  36. 36.
    Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton (1993)Google Scholar
  37. 37.
    Tomas P.: A restriction theorem for the Fourier transform. Bull. Am. Math. Soc. 81, 477–478 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wilcox C.: Theory of Bloch waves. J. Anal. Math. 33, 146–167 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyInstitute for AnalysisKarlsruheGermany

Personalised recommendations