# On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva

Banach Center Publications (1992)

- Volume: 27, Issue: 2, page 295-308
- ISSN: 0137-6934

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topLieberman, Gary. "On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva." Banach Center Publications 27.2 (1992): 295-308. <http://eudml.org/doc/262650>.

@article{Lieberman1992,

author = {Lieberman, Gary},

journal = {Banach Center Publications},

keywords = {quasilinear elliptic equations},

language = {eng},

number = {2},

pages = {295-308},

title = {On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva},

url = {http://eudml.org/doc/262650},

volume = {27},

year = {1992},

}

TY - JOUR

AU - Lieberman, Gary

TI - On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva

JO - Banach Center Publications

PY - 1992

VL - 27

IS - 2

SP - 295

EP - 308

LA - eng

KW - quasilinear elliptic equations

UR - http://eudml.org/doc/262650

ER -

## References

top- [1] L. Boccardo, P. Marcellini and C. Sbordone, ${L}^{\infty}$-regularity for variational problems with sharp non standard growth conditions, Boll. Un. Mat. Ital. (7) 4-A (1990), 219-225. Zbl0711.49058
- [2] H. J. Choe, A regularity theory for a more general class of quasilinear elliptic differential equations and obstacle problems, Arch. Rational Mech. Anal. 114 (1991), 393-394. Zbl0733.35024
- [3] H. J. Choe, Regularity for certain degenerate elliptic double obstacle problems, J. Math. Anal. Appl., to appear. Zbl0798.35057
- [4] E. DiBenedetto, ${C}^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. Zbl0539.35027
- [5] E. DiBenedetto, On the local behavior of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (4) 13 (1986), 487-535. Zbl0635.35052
- [6] E. DiBenedetto and Y.-Z. Chen, On the local behaviour of solutions of singular parabolic equations, Arch. Rational Mech. Anal. 103 (1988), 319-345. Zbl0673.35047
- [7] E. DiBenedetto and Y.-Z. Chen, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131. Zbl0661.35052
- [8] E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, ibid. 349 (1984), 83-128. Zbl0527.35038
- [9] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, ibid. 357 (1985), 1-22. Zbl0549.35061
- [10] E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when 1 < p < 2, Arch. Rational Mech. Anal. 111 (1990), 225-290. Zbl0726.35066
- [11] E. DiBenedetto and N. S. Trudinger, Harnack inequalities for quasi-minima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 295-308. Zbl0565.35012
- [12] T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52-75. Zbl0216.15702
- [13] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248. Zbl0638.49005
- [14] M. Giaquinta and E. Giusti, Global ${C}^{1,\alpha}$ regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math. 351 (1984), 55-65. Zbl0528.35014
- [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin 1983. Zbl0562.35001
- [16] T. Kilpeläinen and W. P. Ziemer, Pointwise regularity of solutions to nonlinear double obstacle problems, Ark. Mat. 29 (1991), 83-106. Zbl0733.35025
- [17] A. G. Korolev, On boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities, Mat. Sb. 180 (1989), 78-100 (in Russian); English transl.: Math. USSR-Sb. 66 (1990), 83-106.
- [18] M. A. Kranosel'skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen 1961.
- [19] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 75-108 (in Russian); English transl.: Math. USSR-Izv. 21 (1984), 67-98.
- [20] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1967.
- [21] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow 1964 (in Russian); English transl.: Academic Press, New York 1968. 2nd Russian ed., 1973.
- [22] G. M. Lieberman, Interior gradient bounds for non-uniformly parabolic equations, Indiana Univ. Math. J. 32 (1983), 579-601. Zbl0491.35021
- [23] G. M. Lieberman, The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Scuola Norm Sup. Pisa (4) 13 (1986), 347-387. Zbl0655.35047
- [24] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. Zbl0675.35042
- [25] G. M. Lieberman, Boundary regularity for solutions of degenerate parabolic equations, ibid. 14 (1990), 501-524. Zbl0703.35098
- [26] 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. Zbl0742.35028
- [27] G. M. Lieberman, Local and boundary regularity for some variational inequalities involving p-Laplaciantype operators, to appear.
- [28] G. M. Lieberman, Regularity of solutions to some degenerate double obstacle problems, Indiana Univ. Math. J. 40 (1991), 1009-1028. Zbl0767.35029
- [29] G. M. Lieberman, Boundary and initial regularity for solutions of degenerate parabolic equations, Nonlinear Anal., to appear. Zbl0782.35036
- [30] J. H. Michael and W. P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), 1427-1448. Zbl0603.49006
- [31] J. H. Michael and W. P. Ziemer, Existence of solutions to obstacle problems, ibid. 17 (1991), 45-71. Zbl0735.35067
- [32] J. Mu, Higher regularity of the solution to the p-Laplacian obstacle problem, J. Differential Equations 95 (1992), 370-384. Zbl0765.49008
- [33] J. Mu and W. P. Ziemer, Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations, Comm. Partial Differential Equations 16 (1991), 821-843. Zbl0742.35010
- [34] L. M. Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), 821-855. Zbl0346.35016
- [35] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. Zbl0488.35017
- [36] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. Zbl0372.35030
- [37] N. N. Ural'tseva, Degenerate quasilinear elliptic systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184-222 (in Russian); English transl.: Sem. Math. V. A. Steklov Math. Inst. Leningrad 7 (1968), 83-99.
- [38] M. Wiegner, On ${C}_{\alpha}$-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Mat. Pura Appl. 145 (1986), 385-405. Zbl0642.35046

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