Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2017, Article ID 5196513, 20 pages
https://doi.org/10.1155/2017/5196513
Research Article

Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions

1Department of Mathematics, Florida International University, Miami, FL 33199, USA
2Department of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, P.O. Box 70377, San Juan, PR 00936-8377, USA

Correspondence should be addressed to Ciprian G. Gal; ude.uif@lagc

Received 2 March 2017; Accepted 30 May 2017; Published 27 June 2017

Academic Editor: Luigi C. Berselli

Copyright © 2017 Ciprian G. Gal and Mahamadi Warma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. G. Gal, M. Grasselli, and A. Miranville, “Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,” in Nonlinear phenomena with energy dissipation, vol. 29 of GAKUTO Internat. Ser. Math. Sci. Appl., pp. 117–139, Gakko Tosho, Tokyo, Japan, 2008. View at Google Scholar · View at MathSciNet
  2. C. G. Gal and A. Miranville, “Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 10, no. 3, pp. 1738–1766, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,” Communications on Pure and Applied Mathematics, vol. 12, pp. 623–727, 1959. View at Publisher · View at Google Scholar · View at MathSciNet
  4. C. G. Gal, G. R. Goldstein, J. A. Goldstein, S. Romanelli, and M. Warma, “Fredholm alternative, semilinear eliptic problems, and Wentzell boundary conditions,” http://arxiv.org/abs/1311.3134.
  5. L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin, Germany, 1976. View at MathSciNet
  6. J. Peetre, “Another approach to elliptic boundary problems,” Communications on Pure and Applied Mathematics, vol. 14, pp. 711–731, 1961. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. I. Vishik, “On general boundary problems for elliptic differential equations,” Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 1, pp. 187–246, 1952. View at Google Scholar · View at MathSciNet
  8. E. M. Landesman and A. C. Lazer, “Nonlinear perturbations of linear elliptic boundary value problems at resonance,” Journal of Applied Mathematics and Mechanics, vol. 19, pp. 609–623, 1970. View at Google Scholar · View at MathSciNet
  9. H. m. Brézis and A. Haraux, “Image d'une somme d'opérateurs monotones et applications,” Israel Journal of Mathematics, vol. 23, no. 2, pp. 165–186, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  10. H. m. Brézis and L. Nirenberg, “Image d'une somme d'opérateurs non linéaires et applications,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 284, no. 21, pp. A1365–A1368, 1977. View at Google Scholar · View at MathSciNet
  11. P. h. Bénilan and M. G. Crandall, “Completely accretive operators,” in Lecture Notes in Pure and Applied Mathematics, Semigroup theory and evolution equations (Delft, 1989), Dekker, New York, NY, USA, 1991. View at Google Scholar
  12. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de hilbert, American Elsevier Publishing Co., Inc., New York, NY, USA, 1973.
  13. G. J. Minty, “Monotone (nonlinear) operators in Hilbert space,” Duke Mathematical Journal, vol. 29, pp. 341–346, 1962. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. G. J. Minty, “On the solvability of nonlinear functional equations of `monotonic'\ type,” Pacific Journal of Mathematics, vol. 14, pp. 249–255, 1964. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1997. View at MathSciNet
  16. J. Necas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, France, 1967. View at MathSciNet
  17. V. G. Maz'ya and S. V. Poborchi, Differentiable functions on bad domains, World Scientific Publishing Co., Inc., 1997. View at MathSciNet
  18. R. A. Adams, Sobolev Spaces. Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  19. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. M. Rao and Z. D. Ren, Applications Of Orlicz Spaces, vol. 250 of Monographs and Textbooks in Pure and Applied Mathematics, CRC Press, Marcel Dekker, Inc., 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. K. Murthy and G. Stampacchia, “Boundary value problems for some degenerate-elliptic operators,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 80, pp. 1–122, 1968. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. V. G. Maz'ya, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, Germany, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  24. H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and optimization, vol. 6 of MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 2006. View at MathSciNet
  25. M. Biegert and M. Warma, “The heat equation with nonlinear generalized Robin boundary conditions,” Journal of Differential Equations, vol. 247, no. 7, pp. 1949–1979, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. Warma, “The ROBin and Wentzell-ROBin Laplacians on LIPschitz domains,” Semigroup Forum, vol. 73, no. 1, pp. 10–30, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. M. Biegert and M. Warma, “Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad” domains,” Advances in Differential Equations, vol. 15, no. 9-10, pp. 893–924, 2010. View at Google Scholar · View at MathSciNet
  28. P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Advanced Texts, Birkhäuser, Basel, Switzerland, 2007. View at MathSciNet