GT ADG-Systèmes Dynamiques
# Dirichlet space over Jordan domains

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#### Orléans institut Denis Poisson

Description

If $U$ be a $C^1$ function with compact support in the plane and $u$ its restriction to the unit circle $\mathbb{S}$, let us denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $\mathbb S$ in the Riemann sphere. About a hundred years ago, Jesse Douglas, for his solution of the Plateau problem, has shown that

\begin{align*}

\iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&= \iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy\\

&= \frac{1}{2\pi}\iint_{\mathbb S\times\mathbb S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|,

\end{align*}

thus giving three ways to express what is called the Dirichlet norm of $u$. The main goal of this talk is to give a characterization of rectifiable Jordan curves for which the obvious analogue of these three (semi-)norms are equivalent.\\

joint work with** Wei Huaying**, Jiangsu Normal University at Xuzhou.