It is shown in this paper that gradient of vector valued function $ u(x), $ solution of a nonlinear elliptic system, cannot be too close to a straight line without $ u(x) $ being regular.
@article{107565,
author = {Josef Dan\v e\v cek and Eugen Viszus},
title = {A note on regular points for solutions of nonlinear elliptic systems},
journal = {Archivum Mathematicum},
volume = {032},
year = {1996},
pages = {105-116},
zbl = {0903.35010},
mrnumber = {1407342},
language = {en},
url = {http://dml.mathdoc.fr/item/107565}
}
Daněček, Josef; Viszus, Eugen. A note on regular points for solutions of nonlinear elliptic systems. Archivum Mathematicum, Tome 032 (1996) pp. 105-116. http://gdmltest.u-ga.fr/item/107565/
Sistemi ellittici in forma divergenza. Regolarita all’interno, Quaderni, Pisa, 1980. (1980) | MR 0668196 | Zbl 0453.35026
On the regularity of weak solutions to nonlinear elliptic systems of second order, Zeitschrift für Analysis und ihre Anwendungen Bd. 9(6) (1990), 535–544. (1990) | MR 1119297
A note one regular points for solutions of elliptic systems, Manuscripta mathematica 29 (1979), 417–426. (1979) | MR 0545051
Function spaces, Academia, Prague, 1977. (1977) | MR 0482102
Introduction to the theory of nonlinear elliptic equations, Teubner-Texte zur Math., Leipzig, 1983. (1983) | MR 0731261
Principio di massimo per i sistemi ellitici quasilineari non diagonali, Bol.U.M.I. (4)6 (1972), 1–10. (1972) | MR 0315281