Limiting Sobolev inequalities and the 1-biharmonic operator

In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L-1. We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L-1-data by Maz'ya (N >= 3) and Brezis-Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L-1, for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber-Krahn type inequalities.