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Tolksdorf, Peter

Everywhere-regularity for some quasilinear systems with a lack of ellipticity. (English) Zbl 0538.35034

Ann. Mat. Pura Appl., IV. Ser. 134, 241-266 (1983).
The author proves the \(C^{1,\alpha}\)-everywhere-regularity of the solutions of quasilinear systems having a similar structure as \(div\{| \nabla u|^{p-2}\nabla u\}=f (1<p<\infty)\). He generalizes results due to K. Uhlenbeck [Acta Math. 138, 219-240 (1977; Zbl 0372.35030)] and P.-A. Ivert [Manuscr. Math. 30, 53-88 (1979; Zbl 0429.35033)]. After a differentiation, the above system is not of diagonal form. Nevertheless, K. Uhlenbeck showed that \(| \nabla u|\) satisfies an inequality similar to \(-\Delta | \nabla u|^ 2+\lambda | \nabla^ 2u|^ 2\leq 0 (\lambda>0)\). With the aid of such an inequality, she established a strong maximum principle for \(\nabla u\) which is the key to the everywhere-regularity. The author gives another rather simple proof of that and he treats more general systems than in the article loc. cit.
Reviewer: J.Frehse

Cited in 1 Review
Cited in 88 Documents

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B50 Maximum principles in context of PDEs

Keywords:

degeneration of ellipticity; everywhere-regularity; quasilinear systems; strong maximum principle

Citations:

Zbl 0372.35030; Zbl 0429.35033
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Full Text: DOI

References:

[1] Almgren, F., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Annals of Mathematics, 87, 321-391 (1968) · Zbl 0162.24703
[2] Caffarelli, L., Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math., 35, 771-832 (1982) · Zbl 0509.35067
[3] Campanato, S., Proprietà di Hölderianità di alcune classi di funzioni, Annali di S.N.S. Pisa, 17, 175-188 (1963) · Zbl 0121.29201
[4] De Giorgi, E., Sulla differentiabilità et l’analiticità delle estremali degli integrali multipli regolari, Memorie delle Acc. Sci. Torino, Ser. 3, 3, 25-43 (1957) · Zbl 0084.31901
[5] E.De Giorgi,Frontiere orientate di misura minima, Sem. Mat. S.N.S., Pisa (1960-61).
[6] De Thelin, F., Local regularity properties for solutions of a nonlinear partial differential equation, Nonlinear Analysis, 6, 839-844 (1982) · Zbl 0493.35021
[7] Di Benedetto, E., C^1+α local regularity of weak solutions of degenerate elliptic equations, Nonl. Anal., 7, 827-850 (1983) · Zbl 0539.35027
[8] Evans, L., A new proof of local C^1,α regularity for solutions of certain degenerate elliptic P.D.E., Journal of Diff. Equations, 45, 356-373 (1982) · Zbl 0508.35036
[9] Gehring, F., The L^p-Integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021
[10] M.GiaquintaMultiple integrals in the calculus of variations and nonlinear elliptic systems, Vorlesungsreihe des SFB72, Universität Bonn, 1981. · Zbl 0498.49001
[11] Giaquinta, M.; Giusti, E., Nonlinear elliptic systems with quadratic growth, Man. Math., 24, 323-349 (1978) · Zbl 0378.35027
[12] Giaquinta, M.; Giusti, E., On the regularity of minima of variational integrals, Acta Math., 148, 31-46 (1982) · Zbl 0494.49031
[13] M.Giaquinta - S.Hildebrandt,Estimation à priori des solutions faibles de certains systèmes non linéaires elliptiques, Séminaire Goulaouic-Meyer-Schwartz (1980-81). · Zbl 0486.35030
[14] Giaquinta, M.; Hildebrandt, S., A priori estimates for harmonic mappings, Journal f.d. reine u. angew. Math., 336, 124-164 (1982) · Zbl 0508.58015
[15] Giaquinta, M.; Modica, G., Regularity results for some classes of higher order non linear elliptic systems, Journal f.d. reine und angew. Math., 311-312, 145-169 (1979) · Zbl 0409.35015
[16] Giusti, E.; Miranda, M., Un esempio di solutioni discontinue per un problema di minima relativo ad un integrale regolare del calcolo delle variazioni, Boll. U.M.I., 2, 1-8 (1968) · Zbl 0155.44501
[17] Giusti, E.; Miranda, M., Sulla regolarità delle soluzioni deboli di una classe di sistemi ellitici quasi-lineari, Arch. Rat. Mech. Anal., 31, 173-184 (1968) · Zbl 0167.10703
[18] Hartman, P.; Stampacchia, G., On some nonlinear elliptic differential-functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102
[19] Hildebrandt, S.; Jost, J.; Widman, K., Harmonic mappings and minimal submanifolds, Inventiones Math., 62, 269-298 (1980) · Zbl 0446.58006
[20] Hildebrandt, S.; Kaul, H.; Widman, K., An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math., 138, 1-16 (1977) · Zbl 0356.53015
[21] Hildebrandt, S.; Widman, K., On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order, Annali di S.N.S. Pisa, 4, 145-178 (1977) · Zbl 0353.35013
[22] Ivert, P., Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung, Man. Math., 30, 53-88 (1979) · Zbl 0429.35033
[23] Ladyzhenskata, O.; Uraltseva, N., Linear and quasilinear elliptic equations (1968), New York: Academic Press, New York · Zbl 0164.13002
[24] J.Lewis,Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. Journal, to appear. · Zbl 0554.35048
[25] Lions, J., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Paris: Dunod, Paris · Zbl 0189.40603
[26] Meier, M., Boundedness and integrability properties of weak solutions of quasilinear elliptic systems, Journal f.d. reine und angew. Math., 333, 191-220 (1982) · Zbl 0476.35029
[27] Morrey, C., Multiple integrals in the calculus of variations (1966), Berlin: Springer-Verlag, Berlin · Zbl 0142.38701
[28] Morrey, C., Partial regularity results for non-linear elliptic systems, Journal of Math. and Mech., 17, 649-670 (1968) · Zbl 0175.11901
[29] Moser, J., On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, 577-591 (1961) · Zbl 0111.09302
[30] J.Nečas,Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity, Theory of Nonlinear Operators, Abh. der Akad. der Wiss. der DDR, 1977. · Zbl 0372.35031
[31] Struwe, M., A counterexample in elliptic regularity theory, Man. Math., 34, 85-92 (1981) · Zbl 0468.35027
[32] P.Tolksdorf,Regularity for a more general class of quasilinear elliptic equations, Journal of Diff. Equations, to appear. · Zbl 0488.35017
[33] P.Tolksdorf,A Strong Maximum Principle and regularity for harmonic mappings, Preprint. · Zbl 0627.58018
[34] Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030
[35] Uraltseva, N., Degenerate quasilinear elliptic systems, Zap. Naucn. Sem. Leningrad. Otel. Mat. Inst. Steklov (LOMI), 7, 184-222 (1968)
[36] Von Wahl, W.; Wiegner, M., Über die Hölderstetigkeit schwacher Lösungen semilinearer elliptischer Systeme mit einseitiger Bedingung, Man. Math., 19, 385-399 (1976) · Zbl 0336.35044
[37] Wiegner, M., Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme, Math. Z., 147, 21-28 (1976) · Zbl 0316.35039
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