transactions of the american mathematical society Volume 277, Number 1, May 1983 CONFORMALLY INVARIANT VARIATIONAL INTEGRALS BY S. GRANLUND, P. LINDQVIST AND O. MARTIO Abstract. Let /: G -» R" be quasiregular and / = / F(x,Vu) dm a conformally invariant variational integral. Holder-continuity, Harnack's inequality and principle are proved for the extremals of /. Obstacle problems and their connection to subextremals are studied. If « is an extremal or a subextremal of /, then u ° / is again an extremal or a subextremal if an appropriate change in F is made. 1. Introduction. A mapping/: G -> R", G open in R", is called quasiregular if /is continuous and ACL" in G with \f'(x) \" < KJ(x, f) a.e. in G for some K > 1. A homeomorphic quasiregular mapping onto fG is called quasiconformal. If n = 2 and K = 1, /= ux + iu2 is analytic or conformai, respectively. The functions ux and u2 are harmonic functions and hence extremals for the Dirichlet-integral. The coordi- nate functions fx,...,f„ of a quasiregular mapping /: G -» R" are extremals of the variational integral /F(x, Vu) dm where F(x, h) »| h |" and F depends on /. It is well known that the Dirichlet-integral remains invariant under conformai mappings/ of the plane domain G, i.e. f \Vu\2dm = f |v(«° f)\2 dm. JfG JG In space and in plane it is possible to rephrase this equality for general kernels F and for quasiconformal mappings when an appropriate change in F is made. For analytic and quasiregular mappings this invariance property has the counterpart which takes into account the many-to-one character of these mappings. The capacity inequalities of quasiconformal and quasiregular mappings, see [G, MRV1 and Ml], form a special case of this invariance property. This phenomenon can also be studied using partial differential equations, the Euler equations of the corresponding variational integrals, and the basic fact is that harmonic functions in plane remain harmonic under composition with analytic functions. Yu. G. ReSetnjak [R4] has studied the corresponding property of quasiregular mappings. A new proof for this result is presented in a more general case. Subharmonic functions form a natural generalization of harmonic functions. The main purpose of this paper is to show that this class of functions has a generalization to the nonhnear case in space as well. We call these functions sub-T'-extremals since for our purposes the extremality property expressed in terms of variational integrals Received by the editors September 9, 1981. 1980 Mathematics Subject Classification. Primary30C70;Secondary 49A21, 35A15, 31B99. Key words and phrases. Variational integrals, quasiregular mappings, subextremals. ©1983 American Mathematical Society 0002-9947/82/0000-0574/$08.50 43 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use