2020/07/29 by Gaven Martin, Cong Yao, Martin, Gaven +1 · 2 citations
Mathematics · #30C62 31A05 49J10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.15149
openalex publication_date 2020/07/29 · openalex created_date 2020/08/03 · openalex updated_date 2026/07/28
We study minimisers of the p-conformal energy functionals, Ep(f):=∫_\ID \IKp(z,f) dz, f|_\IS=f0|_\IS, defined for self mappings f:\ID→\ID with finite distortion and prescribed boundary values f0. Here \IK(z,f) = (‖Df(z)‖2)/(J(z,f)) = (1+|μf(z)|2)/(1-|μf(z)|2) is the pointwise distortion functional and μf(z) is the Beltrami coefficient of f. We show that for quasisymmetric boundary data the limiting regimes p→∞ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for p→1 recovers the harmonic mapping theory. Critical points of Ep always satisfy the inner-variational distributional equation 2p∫_\ID \IKp \fracμf1+|μf|2φ_\zbar dz=∫_\ID \IKp φz dz, ∀φ∈ C0^∞(\ID ). We establish the existence of minimisers in the \em a priori regularity class W1,(2p)/(p+1)(\ID) and show these minimisers have a pseudo-inverse - a continuous W1,2(\ID) surjection of \ID with (h∘ f)(z)=z almost everywhere. We then give a sufficient condition to ensure C∞(\ID) smoothness of solutions to the distributional equation. For instance \IK(z,f)∈ Lrloc(\ID) for any r>p+1 is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further \IK(w,h)∈ L1(\ID) will imply h is a homeomorphism, and together these results yield a diffeomorphic minimiser. We show such higher regularity assumptions to be necessary for critical points of the inner variational equation.