Uniform limits of certain A-harmonic functions with applications to quasiregular mappings
Alexandre Erëmenko, John L. Lewis · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1991
Let u1, 1t2t...ru* be nonconstant uniform limits (on compact subsets) of ,4 harmonic functions in {c : lrl < n} C R' where .4satisfies certain elliptic structure conditions.Theauthorsshowthatifthereexistsl20suchthat(i) {c:u;(n) <-)}n{c:u1@)< -)}=0, (ii)luf-"ll Sl,and(iii)lu,(O)l (),forl<i,,i(rn,thenrn(cwherecdependsonly on thä structure conditions and n.As an application they show that their theorem provides a completely P.D.E.proof of Rickman's generalization of Picard's theorem to quasiregular mappings.1. Introductron Let u: (rr,...,nn) denote apoint in Euclidean n space (R"), andput (*, y) riyi, fr,,U € R', B(r,r):{u:ly-"1 <r}, r)0, r€R'.Let E, 08, and lEl denote the closure, boundary, and Lebesgue n measure of E.lf.g is afunction on R', put M(r, g,to): sup g, gi : mil(g,0) and B(xs,r) t0s 0g 0gt v9: \Arr,5,..., Ar.).Let Io(O), 1 ( p < oo, be the usual space of Lebesgue measurable functions 9 on a domain Q with norm denoted by llgllr.Let W1,o(Q) be the Sobolev space of functional elements with distributional gradients Vg and norm given by llollt,, : llVgllp + llgllr.We say that g € Wl,p(O) locallv, provided s eW1,r(O) whenever O is an open set with O e O.We denote the space of functions with compact support in Q by Co-(O), and set Wr,o(9), equal to the closure in Wt,p(Q) of Co-(Q).For fixed p, I <p < oo, suppose that A : A(t,4) is a function from f,) x R," --+ R' with the following properties: n r / i:1 Supported by