On Conformal, Harmonic Mappings and Dirichlet's Integral

David Kalaj, Miodrag Mateljević

This paper has an expository character, however we present as well some new results and new proofs. We prove a complex version of Dirichlet's principle in the plane and give some applications of it as well as estimates of Dirichlet's integral from below. Some of the results in the plane are generalized to higher dimensions. Roughly speaking, under the appropriate conditions we estimate the $n$-Dirichlet integral of a mapping $u$ defined on a domain $\Omega\subset\Bbb R^n$, $n\geq2$ by the measure of $u(\Omega)$ and show that equality holds if and only if it is injective conformal. Also some sharp inequalities related to the $L^2$ norms of the radial derivatives of vector harmonic mappings from the unit ball in $R^n$, $n\geq2$ are given. As an application, we estimate the 2-Dirichlet integrals of mappings in the Sobolev space $W_1^2$.