Limiting Sobolev inequalities and the 1-biharmonic operator
Résumé
Abstract 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.