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Gradient estimates for semilinear elliptic equations

Published online by Cambridge University Press:  14 November 2011

Gary M. Lieberman
Gary M. Lieberman
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A.and Centre for Mathematical Analysis, Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, Australia
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Synopsis

Estimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 11 - 17
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Gilbarg, D. and Hormander, L.. Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74 (1980), 297–318.CrossRefGoogle Scholar
2Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).Google Scholar
3Ladyzhenskaya, O. A. and Ural'tseva, N. N.. Estimate of the Hölder norm of the solutions of second-order quasilinear elliptic equations of the general form. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 96 (1980), 161'168. English translation in J. Soviet Math. 21 (1983), 762–768.Google Scholar
4Lieberman, G. M.. The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981), 437–497.CrossRefGoogle Scholar
5Lieberman, G. M.. The Dirichlet problem for quasilinear elliptic equationswith continuously differentiable boundary data, to appear.Google Scholar
6Schmidt, K.. Boundary value problems for quasilinear second-order ellipticequations. Nonlinear Anal. 2 (1978), 263–309.CrossRefGoogle Scholar
7Tomi, F.. Über semilineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z. 111 (1969), 350–366.CrossRefGoogle Scholar
8Troianiello, G. M.. Maximal and minimal solutions to a class of elliptic quasilinear problems Proc. Amer. Math. Soc. 94 (1984), 95–101.CrossRefGoogle Scholar