International Journal of Differential Equations

W2,2 A Priori Bounds for a Class of Elliptic Operators

Sara Monsurrò, Maria Salvato, and Maria Transirico

Full-text: Open access

Abstract

We obtain some W2,2 a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 572824, 17 pages.

Dates
Received: 17 June 2011
Accepted: 5 August 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313241

Digital Object Identifier
doi:10.1155/2011/572824

Mathematical Reviews number (MathSciNet)
MR2843509

Zentralblatt MATH identifier
1250.47045

Citation

Monsurrò, Sara; Salvato, Maria; Transirico, Maria. ${W}^{\mathbf{2},\mathbf{2}}$ A Priori Bounds for a Class of Elliptic Operators. Int. J. Differ. Equ. 2011 (2011), Article ID 572824, 17 pages. doi:10.1155/2011/572824. https://projecteuclid.org/euclid.ijde/1485313241


Export citation

References

  • A. Maugeri, D. K. Palagachev, and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, J.Wiley-VCH, New York, NY, USA, 2000.
  • C. Miranda, “Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui,” Annali di Matematica Pura ed Applicata, vol. 63, no. 1, pp. 353–386, 1963.
  • A. Alvino and G. Trombetti, “Second order elliptic equations whose coefficients have their first derivatives weakly-${L}^{n}$,” Annali di Matematica Pura ed Applicata, vol. 138, no. 1, pp. 331–340, 1984.
  • F. Chiarenza and M. Franciosi, “A generalization of a theorem by C. Miranda,” Annali di Matematica Pura ed Applicata, vol. 161, pp. 285–297, 1992.
  • M. Chicco, “Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients,” Annali di Matematica Pura ed Applicata, vol. 92, no. 1, pp. 13–22, 1972.
  • F. Chiarenza, M. Frasca, and P. Longo, “Interior ${W}^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients,” Ricerche di Matematica, vol. 40, no. 1, pp. 149–168, 1991.
  • F. Chiarenza, M. Frasca, and P. Longo, “${W}^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,” Transactions of the American Mathematical Society, vol. 336, no. 2, pp. 841–853, 1993.
  • M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati,” Annali di Matematica Pura ed Applicata, vol. 152, no. 1, pp. 209–226, 1988.
  • A. Canale, P. di Gironimo, and A. Vitolo, “Functions with derivatives in spaces of Morrey type and elliptic equations in unbounded domains,” Studia Mathematica, vol. 128, no. 3, pp. 199–218, 1998.
  • P. di Gironimo and M. Transirico, “Second order elliptic equations in weighted Sobolev spaces on unbounded domains,” Rendiconti della Accademia Nazionale delle Scienze, vol. 15, no. 10, pp. 163–176, 1991.
  • A. Kubica and W. M. Zajaczkowski, “A priori estimates in weighted spaces for solutions of the Poisson and heat equations,” Applicationes Mathematicae, vol. 34, no. 4, pp. 431–444, 2007.
  • L. Sgambati and M. Troisi, “Limitazioni a priori per una classe di problemi ellittici in domini non limitati,” Note di Matematica, vol. 1, no. 2, pp. 225–259, 1981.
  • M. Transirico, M. Troisi, and A. Vitolo, “Spaces of Morrey type and elliptic equations in divergence form on unbounded domains,” Unione Matematica Italiana, vol. 9, no. 1, pp. 153–174, 1995.
  • L. Caso, R. D'Ambrosio, and S. Monsurrò, “Some remarks on spaces of Morrey type,” Abstract and Applied Analysis, vol. 2010, Article ID 242079, 22 pages, 2010.
  • M. Transirico, M. Troisi, and A. Vitolo, “BMO spaces on domains of ${\mathbb{R}}^{n}$,” Ricerche di Matematica, vol. 45, no. 2, pp. 355–378, 1996.
  • L. Caso, P. Cavaliere, and M. Transirico, “A priori bounds for elliptic equations,” Ricerche di Matematica, vol. 51, no. 2, pp. 381–396, 2002.
  • P. Cavaliere, M. Longobardi, and A. Vitolo, “Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains,” Le Matematiche, vol. 51, no. 1, pp. 87–104, 1996.
  • P. Cavaliere and M. Transirico, “The Dirichlet problem for elliptic equations in unbounded domains of the plane,” Journal of Function Spaces and Applications, vol. 8, no. 1, pp. 47–58, 2008.
  • M. Transirico and M. Troisi, “Equazioni ellittiche del secondo ordine a coefficienti discontinui e di tipo variazionale in aperti non limitati,” Unione Matematica Italiana, vol. 2, no. 1, pp. 385–398, 1988.
  • L. Caso, P. Cavaliere, and M. Transirico, “An existence result for elliptic equations with VMO–-coefficients,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1095–1102, 2007.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 2nd edition, 1983.