Dirichlet space over Jordan domains

par Michel Zinsmeister

mardi 21 mai 2024, 14:00 → 15:00 Europe/Paris

Salle de séminaire (Orléans institut Denis Poisson)

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
$$\begin{array}{rl}{\iint }_{\mathbb{D}}|\mathrm{\nabla }{U}_i{|}^2(z)dxdy& ={\iint }_{\overline{\mathbb{C}}\mathrm{\setminus }\overline{\mathbb{D}}}|\mathrm{\nabla }{U}_e{|}^2(z)dxdy\\ & =\frac{1}{2\pi }{\iint }_{\mathbb{S}\times \mathbb{S}}{\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|}^2|d{z}_1||d{z}_2|,\end{array}$$
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.