F-harmonic measure in space
Seppo Granlund, Peter Lindqvist, Олли Мартио · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1982
The use of the harmonic measure is well established in the theory of harmonic and analytic functions.In this paper we present a similar concept which is based on a non-linear Euler equation of the variational integral I r@,Yu)dm where F(x,h)xlhl'.The form of F is essential for our applications in conformal geometry.The purpose of the paper is to show that this concept, called the F'hat' monic measure, is useful even in the non-linear case in space although it has several drawbacks, e.g. it does not define a measure.The paper is a continuation of [GLM] by the same authors and the same nota- tion andlerminology will be used.After constructing the F-harmonic measure in Chapter 2 we show that several classical results of the harmonic measure have anal- ogor, ,trt.-ents for the F-harmonic measure.Among these are Carleman's and plragm6n-Lindelöf's principles.Sets of F-harmonic measure zeto ate considered in Chapter 4 and a simple sufficient metric condition for this is introduced.The con- nection of F-harmonic measures and quasiregular mappings is studied in the last chapter.We prove the invariance of F-harmonic measures under quasiconformal mappings.Note that in this respect the usual harmonic measure is not an invariant, see e.g.[BA], [HP], or even a quasi-invariant.We also present the principle of the F-harmonic measure for quasiregular mappings.These principles include the classical invariance properties of the harmonic measure under conformal and analytic func- tions, respectively.2. Definitions for F-harmonic measure 2.1.Let G be a domain in the n-dimensional Euclidean space R" n>2.Except. in § 3.9, G is assumed to be bounded.We only consider domains which are regular in the following topological sense.2.2.Definition.The domain GcRn is called regular, if no component of its boundary äG reduces to a single point.