The first initial-boundary value problem for quasilinear second order parabolic equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1986)
- Volume: 13, Issue: 3, page 347-387
- ISSN: 0391-173X
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topLieberman, Gary M.. "The first initial-boundary value problem for quasilinear second order parabolic equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 13.3 (1986): 347-387. <http://eudml.org/doc/83983>.
@article{Lieberman1986,
author = {Lieberman, Gary M.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Dirichlet problem; quasilinear; existence; non-existence; smooth initial and boundary values; regularity},
language = {eng},
number = {3},
pages = {347-387},
publisher = {Scuola normale superiore},
title = {The first initial-boundary value problem for quasilinear second order parabolic equations},
url = {http://eudml.org/doc/83983},
volume = {13},
year = {1986},
}
TY - JOUR
AU - Lieberman, Gary M.
TI - The first initial-boundary value problem for quasilinear second order parabolic equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1986
PB - Scuola normale superiore
VL - 13
IS - 3
SP - 347
EP - 387
LA - eng
KW - Dirichlet problem; quasilinear; existence; non-existence; smooth initial and boundary values; regularity
UR - http://eudml.org/doc/83983
ER -
References
top- [1] S. Bernstein, Sur les équations du calcul des variations, Ann. Scuola Norm Sup. Pisa, Cl. Sci., 29 (1912), pp. 431-485. Zbl43.0460.01MR1509153JFM43.0460.01
- [2] D. Edmunds - L. Peletier, Quasilinear parabolic equations, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., (3) 25 (1971), pp. 399-421. Zbl0219.35054MR306704
- [3] A. Friedman, Partial differential equations of parabolic type, Krieger PressMalabar, Florida, 1983.
- [4] Futev, to appear.
- [5] M. Giaquinta - E. Giusti, Global C1,α-regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math., 351 (1984), pp. 55-65. Zbl0528.35014
- [6] D. Gilbarg, Boundary value problems for nonlinear elliptic equations in n variables, in: Nonlinear Problems, pp. 151-159, University of Wisconsin Press, Madison, Wisconsin, 1963. Zbl0121.08401MR146506
- [7] D. Gilbarg - L. Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal., 74 (1980), pp. 297-318. Zbl0454.35022MR588031
- [8] D. Gilbarg - N.S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin -Heidelberg-New York- Tokyo, 1983. Zbl0562.35001MR737190
- [9] E. Giusti, Boundary behavior of nonparametric minimal surface, Indiana Univ. Math. J., 22 (1975), pp. 435-444. Zbl0262.35020MR305253
- [10] M. Gruber, Harnack inequalities for solution of general second order parabolic equations and estimates of their Hölder constants, Math. Z., 185 (1984), pp. 23-43. Zbl0509.35044MR724044
- [11] M. Iannelli - G. Vergara Caffarelli, On the boundary value problem for surfaces of prescribed mean curvature, Sympos. Math., 14 (1974), pp. 473-480. Zbl0307.35044MR388220
- [12] A.V. Ivanov, The first boundary value problem for second order quasi-linear parabolic equations, Zap. Nauchn. Sem. Leningrad, Otdel Mat. Inst. Steklov.(LOMI), 38 (1973), pp. 10-32 [Russian]. English translation in J. Soviet Math., 8 (1977), pp. 354-372. Zbl0404.35054MR374677
- [13] A.V. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order, Trudy Mat. Inst. Steklov., 160 (1982), pp. 1-285 [Russian]. English translation, Proc. Math. Inst. Steklov, 160 (1984), pp. 1-287. Zbl0544.35003MR702172
- [14] H. Jenkins - J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), pp. 170-187. Zbl0159.40204MR222467
- [15] N.V. Krylov, On the maximum principle for nonlinear parabolic and elliptic equations, Izv. Akad. Nauk SSSR, 42 (1978), pp. 1050-1062 [Russian]. English translation in Math. USSR Izv., 13 (1979), pp. 335-347. Zbl0421.35041MR513913
- [16] N.V. Krylov, Boundedly inhomogencous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR, 47 (1983), pp. 75-108 [Russian]. English translation in Math. USSR Izv., 22 (1984), pp. 67-98. Zbl0578.35024MR688919
- [17] N.V. Krylov, On estimates for the derivatives of solutions of nonlinear parabolic equations, Dokl. Akad. Nauk SSSR, 274 (1984), pp. 23-26 [Russian]. English translation in Soviet Math. Dokl., 29 (1984), pp. 14-17. Zbl0598.35057MR730158
- [18] O.A. Ladyzhenskaya - N.N. Ural'tseva, Linear and quasilinear elliptic solutions, Izd. Nauka, Moscow, 1964 [Russian]. English translation, Academic Press, New York, 1968; 2nd Russian ed., 1973.
- [19] O.A. Ladyzhenskaja - V.A. Solonnikov - N.N. Ural'tseva, Linear and quasilinear equations of parabolic type, Izd. Nauka, Moscow, 1967 [Russian]. English translation, Amer. Math. Soc., Providence, R.I., 1968. Zbl0174.15403MR241822
- [20] A. Lichnewsky - R. Temam, Pseudosolutions of the time-dependent minima surface problem, J. Differential Equations, 30 (1978), pp. 340-364. Zbl0368.49016MR521858
- [21] G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary, Comm. Partial Differential Equations, 6 (1981), pp. 437-497. Zbl0458.35039MR612553
- [22] G.M. Lieberman, Interior gradient estimates for nonuniformly parabolic equations, Indiana Univ. Math. J., 32 (1983), pp. 579-601. Zbl0491.35021MR703286
- [23] G.M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), pp. 329-352. Zbl0535.35028MR779924
- [23] G.M. Lieberman, Intermediate Schauder theory for second order parabolic equations. I: Estimates, J. Differential Equations, to appear. Zbl0553.35038MR840590
- [25] G.M. Lieberman, Intermediate Schauder theory for second order parabolic equations. II: Existence, uniqueness and regularity, J. Differential Equations, to appear. Zbl0553.35038MR840591
- [26] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, to appear. Zbl0589.35036MR818099
- [27] G.M. Lieberman - N.S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, to appear. Zbl0619.35047MR833695
- [28] P. Marcellini - K. Miller, Asymptotic growth for the parabolic equations of prescribed mean curvature, J. Differential Equations, 51 (1984), pp. 326-358. Zbl0545.35044MR735204
- [29] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser.A, 264 (1969), pp. 413-495. Zbl0181.38003MR282058
- [30] J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in: Contributions to Nonlinear Functional Analysis, pp. 565-501, Academic Press, New York, 1971. Zbl0271.35004MR402274
- [31] L. Simon, Interior gradient bounds for nonuniformly elliptic equations, Indiana Univ. Math. J., 25 (1976), pp. 821-855. Zbl0346.35016MR412605
- [32] N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math., 21 (1968), pp. 205-226. Zbl0159.39303MR226168
- [33] N.S. Trudinger, The boundary gradient estimate for quasilinear elliptic and parabolic differential equations, Indiana Univ. Math. J., 21 (1972), pp. 657-670. Zbl0236.35022MR289942
- [34] N.S. Trudinger, Fully nonlinear uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc., 278 (1983), pp. 751-769. Zbl0518.35036MR701522
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- Gary Lieberman, On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva
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