Higher regularity and uniqueness for inner variational equations

2020/07/29 by Gaven Martin, Cong Yao, Martin, Gaven +1 · 1 citation
Mathematics · #Analytic and geometric function theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.15150

Abstract

We study local minima of the p-conformal energy functionals, E_{\cal A}^∗(h):=∫_\ID \cal A(\IK(w,h)) J(w,h) dw, h|_\IS=h₀|_\IS, defined for self mappings h:\ID→\ID with finite distortion of the unit disk with prescribed boundary values h₀. Here \IK(w,h) = (‖Dh(w)‖²)/(J(w,h)) is the pointwise distortion functional, and \cal A:[1,∞)→ [1,∞) is convex and increasing with \cal A(t)≈ t^p for some p≥ 1, with additional minor technical conditions. Note \cal A(t)=t is the Dirichlet energy functional. Critical points of E_{\cal A}^∗ satisfy the Ahlfors-Hopf inner-variational equation \cal A'(\IK(w,h)) h_w h_\wbar = Φ where Φ is a holomorphic function. Iwaniec, Kovalev and Onninen established the Lipschitz regularity of critical points. Here we give a sufficient condition to ensure that a local minimum is a diffeomorphic solution to this equation, and that it is unique. This condition is necessarily satisfied by any locally quasiconformal critical point, and is basically the assumption \IK(w,h)∈ L¹(\ID)∩ L^r_loc(\ID) for some r>1.

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• Diffeomorphic solutions of Ahlfors-Hopf equations

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