Calculus of Variations and Geometric Measure Theory

Optimal constants for higher-order functional embeddings

Enea Parini

created by magnani on 07 Nov 2012

13 nov 2012 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University

Abstract.

We consider the problem of finding the optimal constant for the embedding of the space \[ W^{2,1}_\Delta(\Omega) := \left\{ u \in W^{1,1}_0(\Omega)\,
\,\Delta u \in L^1(\Omega)\right\} \] into the space $L^1(\Omega)$, where $\Omega\subset \mathbb R^n$ is a bounded, convex domain, or a bounded domain with boundary of class $C^{1,\alpha}$. This is equivalent to find the first eigenvalue of the 1-biharmonic operator under (generalized) Navier boundary conditions. In this talk we provide an interpretation for the eigenvalue problem, we show some properties of the first eigenfunction, we prove an inequality of Faber-Krahn type, and we compute the first eigenvalue and the associated eigenfunction explicitly for a ball, and in terms of the torsion function for general domains. We will also state some results about the embedding into $L^1(\Omega)$ of the subspace $W^{2,1}_{\Delta,0}(\Omega)$, consisting of those functions of $W^{2,1}_\Delta(\Omega)$ obtained as limits of sequences of smooth functions with compact support.