Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quasilinear elliptic equations with VMO coefficients

Author: Dian K. Palagachev
Journal: Trans. Amer. Math. Soc. 347 (1995), 2481-2493
MSC: Primary 35J65
MathSciNet review: 1308019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Strong solvability and uniqueness in Sobolev space $ {W^{2,n}}(\Omega )$ are proved for the Dirichlet problem

$\displaystyle \left\{ {_{u = \varphi \quad {\text{on}}\partial \Omega .}^{{a^{i... ...{D_{ij}}u + b(x,u,Du) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\Omega }} \right.$

It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory's condition, the $ {a^{ij}}$ are $ V\, M\, O$ functions with respect to $ x$, and structure conditions on $ b$ are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder's fixed point theorem via a priori estimate for the $ {L^{2n}}$-norm of the gradient.

References [Enhancements On Off] (What's this?)

  • [AC] H. Amann and M. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), 779-790. MR 503713 (80a:35047)
  • [Acq] P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl. 161 (1992), 231-270. MR 1174819 (93i:35027)
  • [CFL1] F. Chiarenza, M. Frasca and P. Longo, Interior $ {W^{2,p}}$ estimates for non divergence elliptic equations with discontinuous coefficients, Ricerche Mat. 60 (1991), 149-168. MR 1191890 (93k:35051)
  • [CFL2] -, $ {W^{2,p}}$-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. MR 1088476 (93f:35232)
  • [FK] S. Fučik and A. Kufner, Nonlinear differential equations, Elsevier, New York, 1980.
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [L] G. M. Lieberman, Gradient estimates for semilinear elliptic equations, Proc. Royal Soc. Edinburgh 100A (1985), 11-17. MR 801840 (87a:35080)
  • [M] C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl. 63 (1963), 353-386. MR 0170090 (30:331)
  • [N] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 0109940 (22:823)
  • [Ng] M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207-229. MR 0070014 (16:1116a)
  • [S] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London, Ser. A 264 (1969), 413-496. MR 0282058 (43:7772)
  • [To] F. Tomi, Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 111 (1969), 350-366. MR 0279428 (43:5150)
  • [Tr] G. M. Troianiello, Maximal and minimal subsolutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91 (1984), 95-102. MR 735572 (86a:35057)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J65

Retrieve articles in all journals with MSC: 35J65

Additional Information

Keywords: Strong solution of elliptic PDE, VMO, a priori estimates, Aleksandrov-Pucci maximum principle, Leray-Schauder fixed point theorem
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society