Mappings of smallest mean distortion and free-Lagrangians

Let X,Y ⊂ Rn be bounded domains of the same topological type. We are concerned with mappings f : X → Y, predominately orientation preserving homeomorphisms, in the Sobolev space W 1,p(X,Rn). Thus at almost every x 2 X the linear differential map Df (x) : TxX ≃ Rn -→ TyY ≃ Rn , y = f (x), is represented by the Jacobian matrix Df (x) 2 Rn×n + . Hereafter Rn×n + denotes the space of n × n-matrices with positive determinant.