zbMATH — the first resource for mathematics

Local behavior of solutions of quasilinear elliptic equations with general structure. (English) Zbl 0708.35023
The authors discuss weak solutions of quasilinear equations of the type $-div A(x,u,\nabla u)+B(x,u,\nabla u)=T$ on a domain $$\Omega \subset {\mathbb{R}}^ n$$ where A,B satisfy natural structure and growth conditions, e.g. $A(x,\eta,\xi)\cdot \xi \geq v_ 0(x)| \xi |^ p-v_ 1(x),\quad | A(x,\eta,\xi)| \leq c_ 0(x)| \xi |^{p- 1}+a_ 0(x)$ for some exponent $$1<p<n$$. T is a distribution belonging to some space $$W^{-1,q}(\Omega)$$ but the authors mostly concentrate on the case when T is represented by a nonnegative Radon-measure $$\mu$$ and show that a growth condition of the form $$\mu (B_ r(x))\leq M\cdot r^{n-p+\epsilon}$$ for all balls $$B_ r(x)\subset {\mathbb{R}}^ n$$ is equivalent to the Hölder-continuity of weak solutions.
Reviewer: M.Fuchs

MSC:
 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J60 Nonlinear elliptic equations
Full Text:
References:
 [1] D. R. Adams, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa (3) 25 (1971), 203 – 217. · Zbl 0219.46027 [2] E. DiBenedetto and Neil S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 295 – 308 (English, with French summary). · Zbl 0565.35012 [3] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. · Zbl 0516.49003 [4] Ronald Gariepy and William P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), no. 1, 25 – 39. · Zbl 0389.35023 [5] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001 [6] L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161 – 187. · Zbl 0508.31008 [7] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 – 426. · Zbl 0102.04302 [8] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. · Zbl 0164.13002 [9] Hans Lewy and Guido Stampacchia, On the smoothness of superharmonics which solve a minimum problem, J. Analyse Math. 23 (1970), 227 – 236. · Zbl 0206.40702 [10] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. · Zbl 0189.40603 [11] Norman G. Meyers and Alan Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J. 42 (1975), 121 – 136. · Zbl 0347.35039 [12] J. H. Michael and William P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), no. 12, 1427 – 1448. · Zbl 0603.49006 [13] B. Michaux, J.-M. Rakotoson, and J. Shen, On the existence and regularity of solutions of a quasilinear mixed equation of Leray-Lions type, Acta Appl. Math. 12 (1988), no. 3, 287 – 316. · Zbl 0669.35080 [14] -, On the approximation of a quasilinear mixed problem, Preprint, Institute for Appl. Math. and Sci. Comp., #8808, 1988, MMAN (to appear). [15] Jean-Michel Rakotoson, Réarrangement relatif dans les équations elliptiques quasi-linéaires avec un second membre distribution: application à un théorème d’existence et de régularité, J. Differential Equations 66 (1987), no. 3, 391 – 419 (French, with English summary). · Zbl 0652.35041 [16] J.-M. Rakotoson and R. Temam, Relative rearrangement in quasilinear elliptic variational inequalities, Indiana Univ. Math. J. 36 (1987), no. 4, 757 – 810. [17] James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247 – 302. · Zbl 0128.09101 [18] Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721 – 747. · Zbl 0153.42703
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.