Gradient estimates for a new class of degenerate elliptic and parabolic equations

Lieberman, Gary M.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 21 (1994) no. 4, pp. 497-522.

[1] H.J. Choe, Interior behaviour of minimizers for certain functionals with nonstandard growth. Nonlinear Anal. 19 (1992), 933-945. | MR | Zbl

[2] E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Norm. Sup. Cl. Sci. 13 (1986), 487-535. | Numdam | MR | Zbl

[3] M. Giaquinta, Growth conditions and regularity a counterexample. Manuscripta Math. 50 (1987), 245-248. | MR | Zbl

[4] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin -Heidelberg-New York- Tokyo, 1983. | MR | Zbl

[5] O.A. Ladyzhenskaya - N.N. Utal'Tseva, Linear and Quasilinear Elliptic Equations, "Nauka", Moscow, 1964 (Russian); English transl., 2nd Russian ed., 1973, Academic Press, New York, 1968. | MR | Zbl

[6] G.M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), 218-257. | MR | Zbl

[7] G.M. Lieberman, Interior gradient bounds for non-uniformly parabolic equations, Indiana Univ. Math. J. 32 (1983), 579-601. | MR | Zbl

[8] G.M. Lieberman, The first initial-boundary value problem for quasilinear second order parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13 (1986), 347-387. | Numdam | MR | Zbl

[9] G.M. Lieberman, The conormal derivative problem for non-uniformly parabolic equations. Indiana Univ. Math. J. 37 (1988), 23-73; Addenda, ibid. 39 (1990), 270-281. | MR | Zbl

[10] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations. Comm. Partial Differential Equations 16 (1991), 311-361. | MR | Zbl

[11] G.M. Lieberman, The conormal derivative problem for equations of variational type in nonsmooth domains, Trans. Amer. Math. Soc. 330 (1992), 41-67. | MR | Zbl

[12] G.M. Lieberman, Gradient estimates for a class of elliptic systems, Ann. Mat. Pura Appl. (to appear). | MR | Zbl

[13] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Rational Mech. Anal. 105 (1989), 267-284. | MR | Zbl

[14] P. Marcellini, Regularity of existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), 1-30. | MR | Zbl

[15] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), 296-333. | MR | Zbl

[16] J.H. Michael - L.M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of Rn, Comm. Pure Appl. Math. 26 (1973), 361-379. | MR | Zbl

[17] P.Z. Mkrtychyan, Singular quasilinear parabolic equation arising in nonstationary filtration theory, Izv. Akad. Nau. Armyan. SSSR. Mat. 24 (1989), 103-116; English transl. in Soviet J. Contemp. Math. 24 (1989), 1-13. | MR | Zbl

[18] P.Z. Mkrtychyan, An estimate of the solution gradient and the classical solvability of the first initial-boundary value problem for a class of quasilinear nonuniformly parabolic equations, Izv. Akad. Nauk Armyan. SSR. Ser. Mat. 24 (1989), 293-299; English transl. in Soviet J. Contemp. Math. 24 (1989), 85-91. | MR | Zbl

[19] J.B. Serrin, Gradient estimates for solutions of nonlinear eliptic and parabolic equations. In: Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 565-701. | MR | Zbl

[20] L. Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), 821-855. | MR | Zbl

[21] G.M. Troianiello, Maximal and minimal solutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91 (1984), 95-101. | MR | Zbl