Jani Onninen

Extremal mappings of finite distortion

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that

Mappings of finite distortion: Capacity and modulus inequalities

f_o

Mappings of finite distortion: Sharp Orlicz-conditions

is a homeomorphism of the circle with

The Nitsche conjecture

f_{o}^{-1} \in {\cal W}^{1/2, 2}

Hopf Differentials and Smoothing Sobolev Homeomorphisms

. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on

Diffeomorphic Approximation of Sobolev Homeomorphisms

f_o

Existence of Energy-Minimal Diffeomorphisms Between Doubly Connected Domains

is sharp. Classically the mapping

Mappings of finite distortion: The sharp modulus of continuity

f_o

Estimates of Jacobians by subdeterminants

is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values

Quasihyperbolic boundary conditions and capacity: Holder continuity of quasiconformal mappings

f_o