演講訊息 TALKS

Extremal problems for quasiconformal mappings typically involve two domains Omega1 and Omega2 in n-space, (or two Riemann surfaces) for which there exists a quasiconformal mapping f:Omega1 --> Omega2, and ask for a quasiconformal map u:Omega1--> Omega2 which minimizes one or more of the dilation functions in a given class of competitors which are usually other quasiconformal mappings with same boundary data as f on a portion (or all) of the boundary of the domain Omega1 or in the same homotopy class as the given map f. In a joint work with Andrew Raich (UARK) we study this L^{\infty} variational problem from a PDE point of view which echoes some of G. Aronsson's work in the 60's, concerning absolute minimal extensions and the infinity Laplacian. We derive a non-linear, degenerate elliptic, PDE system as a formal limit of Euler Lagrange equations for L^p approximations for the dilation functional and we prove that solving such a system provides necessary and sufficient conditions for local forms of extremality of C^2 quasiconformal diffeomorphisms. We also study a family of gradient flows for L^p dilation and prove short time existence of solutions for smooth data.