2022/07/13 by Martin, Gaven, Yao, Cong · 2 citations
#30C62 31A05 49J10 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.05935
For an arbitrary convex function Ψ:[1,∞) → [1,∞), we consider uniqueness in the following two related extremal problems: Problem A boundary value problem: Establish the existence of, and describe the mapping f, achieving inff \ ∫\Bbb D Ψ(\Bbb K(z,f)) dz : f:\Bbb D → \Bbb D \mboxa homeomorphism in W1,10(\Bbb D)+f0 \. Here the data f0:\Bbb D → \Bbb D is a homeomorphism of finite distortion with ∫\Bbb D Ψ(\Bbb K(z,f0)) dz<∞ -- a barrier. Next, given two homeomorphic Riemann surfaces R and S and data f0:R → S a diffeomorphism. \noindent\bf Problem B \em (extremal in homotopy class): Establish the existence of, and describe the mapping f, achieving inff \ ∫R Ψ(\Bbb K(z,f)) dσ(z) : f a homeomorphism homotopic to f0 \. There are two basic obstructions to existence and regularity. These are first, the existence of an Ahlfors-Hopf differential and second that the minimiser is a homeomorphism. When these restrictions are met (as they often can be) we show uniqueness is assured. These results are established through a generalisation the classical Reich-Strebel inequalities to this variational setting.