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International Journal of Differential Equations
Volume 2011, Article ID 453727, 18 pages
http://dx.doi.org/10.1155/2011/453727
Research Article

Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity

Department of Mathematics, University of Salerno, 84084 Fisciano, Italy

Received 4 May 2011; Accepted 8 September 2011

Academic Editor: Julio Rossi

Copyright © 2011 G. Galise and A. Vitolo. 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. H. Brezis, “Semilinear equations in n without condition at infinity,” Applied Mathematics and Optimization, vol. 12, no. 3, pp. 271–282, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. J. Esteban, P. L. Felmer, and A. Quaas, “Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data,” Proceedings of the Edinburgh Mathematical Society, vol. 53, no. 1, pp. 125–141, 2010. View at Publisher · View at Google Scholar
  3. L. Escauriaza, “W2,n a priori estimates for solutions to fully nonlinear equations,” Indiana University Mathematics Journal, vol. 42, no. 2, pp. 413–423, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. G. Crandall and A. Swiech, “A note on generalized maximum principles for elliptic and parabolic PDE,” in Evolution Equations: Proceedings in Honor of J.A. Goldstein's 60th Birthday, vol. 234 of Lecture Notes in Pure and Appl. Math., pp. 121–127, Dekker, New York, NY, USA, 2003. View at Google Scholar · View at Zentralblatt MATH
  5. S. Koike and A. Swiech, “Maximum principle and existence of Lp viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms,” Nonlinear Differential Equations and Applications, vol. 11, no. 4, pp. 491–509, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Da Lio and B. Sirakov, “Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations,” Journal of the European Mathematical Society, vol. 9, no. 2, pp. 317–330, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. L. A. Caffarelli, “Interior a priori estimates for solutions of fully nonlinear equations,” Annals of Mathematics, vol. 130, no. 1, pp. 189–213, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Niki Winter, “W2,p and W1,p-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 28, no. 2, pp. 129–164, 2009. View at Publisher · View at Google Scholar
  9. A. Swiech, “W1,p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations,” Advances in Differential Equations, vol. 2, no. 6, pp. 1005–1027, 1997. View at Google Scholar · View at Zentralblatt MATH
  10. X. Cabré and L. A. Caffarelli, “Interior C2,α-regularity theory for a class of nonconvex fully nonlinear elliptic equations,” Journal de Mathématiques Pures et Appliquées, vol. 82, no. 5, pp. 573–612, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. G. Crandall, M. Kocan, P. L. Lions, and A. Swiech, “Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations,” Electronic Journal of Differential Equations, vol. 24, pp. 1–20, 1999. View at Google Scholar · View at Zentralblatt MATH
  12. H. Ishii, “On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs,” Communications on Pure and Applied Mathematics, vol. 42, no. 1, pp. 15–45, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. A. Labutin, “Removable singularities for fully nonlinear elliptic equations,” Archive for Rational Mechanics and Analysis, vol. 155, no. 3, pp. 201–214, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, vol. 43 of Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1995.
  15. L. Caffarelli, M. G. Crandall, M. Kocan, and A. Swiech, “On viscosity solutions of fully nonlinear equations with measurable ingredients,” Communications on Pure and Applied Mathematics, vol. 49, no. 4, pp. 365–397, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. M. G. Crandall, H. Ishii, and P. L. Lions, “User's guide to viscosity solutions of second order partial differential equations,” American Mathematical Society. Bulletin, vol. 27, no. 1, pp. 1–67, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, vol. 13 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, Japan, 2004.
  18. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer, Berlin, Germany, 1983.
  19. B. Sirakov, “Solvability of uniformly elliptic fully nonlinear PDE,” Archive for Rational Mechanics and Analysis, vol. 195, no. 2, pp. 579–607, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, Fla, USA, 1992.
  21. I. Capuzzo Dolcetta and A. Vitolo, “Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations,” Discrete and Continuous Dynamical Systems A, vol. 28, no. 2, pp. 539–557, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH