The Lp Teichmüller Theory: Existence and Regularity of Critical Points.

We study minimisers of the p-conformal energy functionals, E p (f) : = ∫ D K p (z , f) d z , f | S = f 0 | S , defined for self mappings f : D → D with finite distortion and prescribed boundary values f 0 . Here K (z , f) = ‖ D f (z) ‖ 2 J (z , f) = 1 + | μ f (z) | 2 1 - | μ f (z) | 2 is the pointwise distortion functional and μ f (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 E p always satisfy the inner-variational distributional equation 2 p ∫ D K p μ f ¯ 1 + | μ f | 2 φ z ¯ d z = ∫ D K p φ z d z , ∀ φ ∈ C 0 ∞ (D). We establish the existence of minimisers in the a priori regularity class W 1 , 2 p p + 1 (D) and show these minimisers have a pseudo-inverse - a continuous W 1 , 2 (D) surjection of D with (h ∘ f) (z) = z almost everywhere. We then give a sufficient condition to ensure C ∞ (D) smoothness of solutions to the distributional equation. For instance K (z , f) ∈ L loc p + 1 (D) is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further K (w , h) ∈ L 1 (D) 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. [ABSTRACT FROM AUTHOR]