Parini: Optimal constants for higher-order functional embeddings
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. http://cvgmt.sns.it/seminar/301/
When
Tue Nov 13, 2012 4pm – 5pm Coordinated Universal Time
Where
Sala Seminari, Department of Mathematics, Pisa University (map)
