17 nov 2005

Abstract.

Cari Amici,

Gioved{\'\i} 17 Novembre, alle ore 17:30, nella Sala dei Seminari del Dipartimento di Matematica, Sven Winklmann, dell'Universitaet Duisburg-Essen e attualmente al Centro De Giorgi, comincerà il ciclo di seminari di Calcolo delle Variazioni parlando di:

"Curvature estimates and Bernstein type results for parametric variational problems"

Segue l'estratto della presentazione.

In this talk we consider immersed hypersurfaces in euclidean $(n+1)$-space which are stationary with respect to an elliptic parametric functional with integrand $F=F(N)$ depending on normal directions. We derive a generalized Simons' inequality for the Laplacian of the length of the anisotropic second fundamental form with respect to a perturbed metric associated with $F$. As a consequence, we show that curvature estimates leading to a Bernstein type result for stable hypersurfaces of dimension $n \leq 5$ can be proved if $F$ is $C^3$-close to the area integrand. In particular, this extends the well-known curvature estimates of Schoen-Simon-Yau for stable minimal hypersurfaces as well as Simon's estimate for $F$-minimizers. Finally, we briefly discuss the application of our estimates to the corresponding non-parametric problem.