Quasilinear elliptic equations with VMO coefficients
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- by Dian K. Palagachev PDF
- Trans. Amer. Math. Soc. 347 (1995), 2481-2493 Request permission
Abstract:
Strong solvability and uniqueness in Sobolev space ${W^{2,n}}(\Omega )$ are proved for the Dirichlet problem \[ \left \{ {_{u = \varphi \quad {\text {on}}\partial \Omega .}^{{a^{ij}}(x,u){D_{ij}}u + b(x,u,Du) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\Omega }} \right .\] 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
- Herbert Amann and Michael G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 779–790. MR 503713, DOI 10.1512/iumj.1978.27.27050
- Paolo Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl. (4) 161 (1992), 231–269. MR 1174819, DOI 10.1007/BF01759640
- Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168. MR 1191890
- Filippo Chiarenza, Michele Frasca, and Placido Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841–853. MR 1088476, DOI 10.1090/S0002-9947-1993-1088476-1 S. Fučik and A. Kufner, Nonlinear differential equations, Elsevier, New York, 1980.
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Gary M. Lieberman, Gradient estimates for semilinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), no. 1-2, 11–17. MR 801840, DOI 10.1017/S0308210500013597
- Carlo Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl. (4) 63 (1963), 353–386 (Italian). MR 170090, DOI 10.1007/BF02412185
- L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
- Mitio Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207–229. MR 70014
- 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 282058, DOI 10.1098/rsta.1969.0033
- Friedrich Tomi, Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 111 (1969), 350–366 (German). MR 279428, DOI 10.1007/BF01110746
- Giovanni M. Troianiello, Maximal and minimal solutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91 (1984), no. 1, 95–101. MR 735572, DOI 10.1090/S0002-9939-1984-0735572-1
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 2481-2493
- MSC: Primary 35J65
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308019-6
- MathSciNet review: 1308019