×

Gradient estimates for a class of elliptic systems. (English) Zbl 0819.35019

The author proves partial regularity and everywhere regularity for solutions of a second order nonlinear elliptic system of the form \[ D_ \alpha (A_ i^ \alpha (x,u,Du))+ B_ i (x,u, Du)=0 \] in \(\Omega\) under additional structure conditions imposed on coefficients. Denote \((G^{\alpha \beta})\), \((g_{ij})\) positive matrices on \(\Omega\times \mathbb{R}^ N\) and set \(v= (G^{\alpha \beta} (x,z) g_{ij} (x,z) p_ \alpha^ i p^ \beta_ j )^{1/2}\). Then the main assumption is that there is a scalar function \(F\) on \(\Omega \times \mathbb{R}^ N\times \mathbb{R}^{Nn}\) satisfying \(A_ i^ \alpha (x,z,p)= \partial F(v)/ \partial p_ \alpha^ i\). The proof is based on a modification of Moser’s iteration scheme.
Reviewer: J.Stará (Praha)

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bombieri, E.; De Giorgi, E.; Miranda, M., Una maggiorazione a priori relativa alle ipersurfici minimali non parametriche, Arch. Rational Mech. Anal., 32, 255-267 (1969) · Zbl 0184.32803
[2] Fusco, N.; Hutchinson, J., Partial regularity for minimisers of certain functionals having non quadratic growth, Ann. Mat. Pura Appl., 155, 1-24 (1989) · Zbl 0698.49001
[3] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Equations, Ann. Math. Study,105 (1983), Princeton, N.J.: Princeton University Press, Princeton, N.J.
[4] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, Berlin-Heidelberg-New York-Tokyo · Zbl 0562.35001
[5] Ladyzhenskaya, O. A.; Ural’Tseva, N. N., Linear and Quasinlinear Elliptic Equations (1964), Moscow: Izdat. «Nauka», Moscow · Zbl 0143.33602
[6] Lieberman, G. M., The conormal derivative problem for elliptic equations of variational type, J. Diff. Eqs., 49, 218-257 (1983) · Zbl 0476.35032
[7] Lieberman, G. M., The conormal derivative problem for non-uniformly parabolic equations, Indiana Univ. Math. J., 32, 23-72 (1988) · Zbl 0707.35077
[8] Meier, M., Boundedness and integrability properties of weak solutions of quasilinear elliptic systems, J. Reine Angew. Math., 333, 191-220 (1982) · Zbl 0476.35029
[9] Michael, J. H.; Simon, L. M., Sobolev and mean-value inequalities on generalized sub- manifolds of R^n, Comm. Pure Appl. Math., 26, 361-379 (1973) · Zbl 0256.53006
[10] Moser, J., A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13, 457-468 (1960) · Zbl 0111.09301
[11] Simon, L. M., Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J., 25, 827-855 (1976)
[12] Tolksdorf, P., Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134, 241-266 (1983) · Zbl 0538.35034
[13] Uhlenbeck, L., Regularity for a class of non-linear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.