Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:01:17.931Z Has data issue: false hasContentIssue false

On an estimate for solutions of nonlinear elliptic variational inequalities1)

Published online by Cambridge University Press:  22 January 2016

Haruo Nagase*
Affiliation:
Suzuka College of Technology, Suzuka, 510-02 Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Ω be a bounded domain in Rn with the boundary of class C0,1 and E be a compact subset (resp. a compact subset on an (n —dimensional hypersurface of class C0,1) in Ω. We assume that the usual function spaces and are known.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

Footnotes

1)

The content of this paper was lectured by the author at the Nonlinear P.D.E. Symposium, held at Tokyo University from February 24 to 26, 1986.

References

Bibliogrphy

[1] Arkhipova, A. A., A problem with discontinuous obstruction for uniformly elliptic equation, Vestn. Lenigr. Univ., 19 (1974), 154155.Google Scholar
[2] Arkhipova, A. A., On limiting smoothness of the solution of a problem with a two-side barrier, Vestn. Lenigr. Univ. Mat., 17 (1984), 59.Google Scholar
[3] Brézis, H.-Kinderlehrer, D., The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 17 (1974), 831844.CrossRefGoogle Scholar
[4] Stampacchia, G., Sur la régularité de la solution d’inéquations elliptiques, Bull. Soc. Math. France, 96 (1968), 153180.Google Scholar
[5] Chang, K. C., The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math., 38 (1980), 117146.CrossRefGoogle Scholar
[6] Cooper, F., Existence and regularity of solutions with bounded gradient of variational inequalities of a certain class, Quart. J. Math. Oxford, 26 (1975), 203214.Google Scholar
[7] Dias, J. P., Une classe de problèmes variationnels non linéarires de type elliptique ou parabolique, Ann. Mat. Pura Appl., 92 (1972), 263322.Google Scholar
[8] DiBenedetto, E., C1,α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. T.M.A., 7 (1983), 827850.Google Scholar
[9] Domarkas, A., Regularity of solutions of quasilinear elliptic equations with unilateral boundary conditions, Lithu. Math. J., 20 (1980), 813.Google Scholar
[10] Domarkas, A., Unilateral problems for quasilinear elliptic equation, ibid., 21 (1981), 317327.Google Scholar
[11] Evans, L. C., A new proof of local C1,α regularity for solutions of certain degenerate elliptic P.D.E., J. Differential Equations, 45 (1982), 356373.Google Scholar
[12] Frehse, J., Two dimensional variational problems with thin obstacles, Math. Z., 143 (1975), 279288.CrossRefGoogle Scholar
[13] Frehse, J., On Signorini’s problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup. Pisa, 4 (1977), 343362.Google Scholar
[14] Frehse, J., On the smoothness of solutions of variational inequalities with obstacles, Banach Center Publ., 10 (1983), 87128.CrossRefGoogle Scholar
[15] Frehse, J., Mosco, U., Irregular obstacles and quasi-variational inequalities of stochastic impulse control, Ann. Scuola Norm. Sup. Pisa, 9 (1982), 105157.Google Scholar
[16] Garroni, M. G., Regularity of a nonlinear variational inequality with obstacle on the boundary, B.U. M.I. Anal. Fun. Appl. Suppl., 1 (1980), 267286.Google Scholar
[17] Gerhardt, C., Regularity of solutions of nonlinear variational inequalities, Arch. Rational Mech. Anal., 52 (1973), 389393.Google Scholar
[18] Gerhardt, C., Global C1,1-regularity for solutions of quasilinear variational inequalities, ibid., 89 (1985), 8392.Google Scholar
[19] Giaquinta, M.-Modica, G., Regolarità Lipschtziana per la soluzione di alcuni problemi di minimo con vincola, Ann. Mat. Pura Appl., 106 (1975), 95117.CrossRefGoogle Scholar
[20] Hayasida, K.-Nagase, H., On solutions of variational inequalities constrained on a subset of positive capacity, Nagoya Math. J., 97 (1985), 5169.CrossRefGoogle Scholar
[21] Huisken, G., C1,1 regularity of solutions to variational inequalities, Proc. Centre Math. Anal. Austral. Nat. Univ., 5 (1984), 8590.Google Scholar
[22] Kinderlehrer, D., The regularity of the solution to a certain variational inequality, Proc. Symp. Pure Appl. Math., 23, A. M. S. Providence.Google Scholar
[23] Kinderlehrer, D., Variational inequalities with lower dimensional obstacles, Israel J. Math., 16 (1971), 339348.CrossRefGoogle Scholar
[24] Kinderlehrer, D., The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl., 60 (1980), 193212.Google Scholar
[25] Kuttler, K. L. Jr., Degenerate variational inequalities of evolution, Nonlinear Anal. T.M.A., 8 (1984), 837850.Google Scholar
[26] Lewis, J. L., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849858.CrossRefGoogle Scholar
[27] Lewy, H.-Stampacchia, G., On existence and smoothness of solutions of some noncoercive variational inequalities, Arch. Rational Mech. Anal., 41 (1971), 241252.CrossRefGoogle Scholar
[28] Lindqvist, P., On the growth of the solutions of the differential equation div. in n-dim. space, J. Differential Equations, 58 (1985), 307317.CrossRefGoogle Scholar
[29] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites nonlinéaries, Dunod Gautyier-Villars, 1969.Google Scholar
[30] Michel, T. H., Regularity for solutions to obstacle problems, Proc. Centre, Math. Anal. Austral. Nat. Univ., 8 (1984), 2937.Google Scholar
[31] Schaaf, R., Regularity of solutions to linear and quasilinear variational inequalities with two obstacles, Analysis, 2 (1982), 337345.Google Scholar
[32] Stampacchia, G., Regularity of solutions of some variational inequalities Proc. Symp. Pure Math., 18, A. M. S. Providence.Google Scholar
[33] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equation, J. Differential Equations, 51 (1984), 126150.Google Scholar
[34] Ural’tseva, N. N., Regularity of the solutions of variational inequalities, J. Soviet Math., 3 (1975), 565573.CrossRefGoogle Scholar
[35] Beirão da Veiga, H., Proprietà di sommabilità e di limitatezza per soluzioni di disequazioni variazionali ellittiche, Rend. Sem. Nat. Padova, 46 (1971), 141171.Google Scholar
[36] Beirão da Veiga, H., Sur la régularité des solutions de l’équation div. avec des conditions aux limites unilatérales et mêlées, Ann. Mat. Pura Appl., 93 (1972), 173230.Google Scholar
[37] Beirão da Veiga, H., Conti, F., Equazioni ellittiche non lineari con ostacoli sottili applicazioni allo studio dei punti regolari, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 533562.Google Scholar
[38] Williams, G. H., Lipschitz continuous solutions for nonlinear obstacle problems, Math. Z., 154 (1977), 5165.Google Scholar
[39] Williams, G. H., Smooth solutions for variational inequalities with obstacles, Indiana Univ. Math. J., 31 (1982), 317325.Google Scholar
[40] Yakovlev, G. H., A variational problem, Differential Equations, 5 (1969), 960966.Google Scholar
[41] Yakovlev, G. H., The first boundary value problem for quasilinear elliptic equations of second order, Proc. Steklov Inst. Math., 117 (1972), 381403.Google Scholar
[42] Yakovlev, G. H., Properties of solutions of a class of second-order quasilinear elliptic equations in divergence form, ibid., 131 (1974), 242252.Google Scholar
[43] Yakovlev, G. H., Some properties of solutions of quasilinear elliptic equations, ibid., 134 (1975), 441458.Google Scholar
[44] Yosida, K., Functional analysis, Springer-Verlag, 1968.Google Scholar