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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Solvability Conditions For Some Non Fredholm Operators in a Layer in Four Dimensions

Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 59--71 | DOI:10.5890/DNC.2014.03.005

Vitaly Volpert$^{1}$; Vitali Vougalter$^{2}$

$^{1}$ Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France

$^{2}$ Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch 7701, South Africa

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Abstract

We study solvability in H2 of certain linear nonhomogeneous elliptic prob- lems involving the sum of the periodic Laplacian and a Schrödinger oper- ator without Fredholm property and prove that under reasonable technical conditions the convergence in L2 of their right sides implies the existence and the convergence in H2 of the solutions. We generalize the methods of spectral and scattering theory for Schro ̈dinger type operators from our preceding work [1].

References

  1. [1]  Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non Fredholm operators, Proceedings of the Edinburgh Mathematical Society (2), 54 (1), 249-271.
  2. [2]  Volpert, V., Kazmierczak, B., Massot, M., and Peradzynski, Z. (2002), Solvability conditions for elliptic problems with non-Fredholm operators, Applicationes Mathematicae, 29(2), 219-238.
  3. [3]  Vougalter, V. and Volpert, V. (2010), On the solvability conditions for some non Fredholm operators, International Journal of Pure and Applied Mathematics, 60(2), 169-191.
  4. [4]  Vougalter, V. and Volpert, V. (2012), On the solvability conditions for the diffusion equation with convection terms, Communications on Pure and Applied Analysis, 11(1), 365-373.
  5. [5]  Vougalter, V. and Volpert, V.(2010), Solvability relations for some non Fredholm operators, International Electronic Journal of Pure and Applied Mathematics, 2(1), 75-83.
  6. [6]  Volpert, V. and Vougalter, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1-9.
  7. [7]  Vougalter, V. and Volpert, V. (2012), Solvability conditions for a linearized Cahn-Hilliard equation of sixth order, Mathematical Modelling of Natural Phenomena, 7(2), 146-154.
  8. [8]  Vougalter, V. and Volpert, V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Analysis and Mathematical Physics, 2(4), 473-496.
  9. [9]  Benkirane, N. (1988), Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans Rn, CRAS, 307, Série I, 577-580.
  10. [10]  Ducrot, A., Marion, M., and Volpert, V. (2005), Systemes de réaction-diffusion sans propriété de Fredholm, CRAS, 340, 659-664.
  11. [11]  Ducrot, A., Marion, M., and Volpert, V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances in Difference Equations, 13(11-12), 1151-1192.
  12. [12]  Amrouche, C., Girault, V., and Giroire, J. (1997), Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, Journal deMathématiques Pures et Appliquées, 76 , 55-81.
  13. [13]  Amrouche, C. and Bonzom, F. (2008), Mixed exterior Laplace's problem, Journal of Mathematical Analysis and Applications, 338 , 124-140.
  14. [14]  Bolley, P. and Pham, T.L. (1993), Propriété d'indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans Rn, Journal de Mathématiques Pures et Appliquées , 72 105-119.
  15. [15]  Bolley, P. and Pham, T.L. (2001), Propriété d'indice en théorie Hölderienne pour le problème extérieur de Dirichlet, Communications in Partial Differential Equations, 26(1-2), 315-334.
  16. [16]  Volpert, V. (2011), Elliptic partial differential equations. Volume 1. Fredholm theory of elliptic problems in unbounded domains, Birkhauser,
  17. [17]  Volpert, V. and Vougalter, V. (2013), Solvability in the sense of sequences to some non-Fredholm operators, Electronic Journal of Differential Equations , 2013(16), 1-16.
  18. [18]  Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Documenta Mathematica, 16, 561-580.
  19. [19]  Jonsson, B.L.G., Merkli, M., Sigal, I.M., and Ting,F. Applied Analysis, In preparation.
  20. [20]  Lieb,E. and Loss, M. (1997), Analysis. Graduate Studies in Mathematics, 14, American Mathematical Society, Providence .
  21. [21]  Kato, T. (1965/1966)Wave operators and similarity for some non-selfadjoint operators,Mathematische Annalen, 162 , 258-279.
  22. [22]  Rodnianski, I. and Schlag, W. (2004), Time decay for solutions of Schrödinger equations with rough and timedependent potentials, Inventiones mathematicae, 155(3), 451-513.
  23. [23]  Reed, M. and Simon, B.(1979), Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, New York.
  24. [24]  Cycon, H.L., Froese, R.G., Kirsch, W., and Simon, B. (1987), Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin .
  25. [25]  Goldberg, M. and Schlag, W. (2004), A limiting absorption principle for the three-dimensional Schrödinger equation with Lp potentials, International Mathematics Research Notices , 75, 4049-4071.

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