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Gradient estimates for semilinear elliptic equations

  • Gary M. Lieberman (a1)


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.



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1Gilbarg, D. and Hormander, L.. Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74 (1980), 297318.
2Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).
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.
4Lieberman, G. M.. The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981), 437497.
5Lieberman, G. M.. The Dirichlet problem for quasilinear elliptic equationswith continuously differentiable boundary data, to appear.
6Schmidt, K.. Boundary value problems for quasilinear second-order ellipticequations. Nonlinear Anal. 2 (1978), 263309.
7Tomi, F.. Über semilineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z. 111 (1969), 350366.
8Troianiello, G. M.. Maximal and minimal solutions to a class of elliptic quasilinear problems Proc. Amer. Math. Soc. 94 (1984), 95101.

Gradient estimates for semilinear elliptic equations

  • Gary M. Lieberman (a1)


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