Limiting Sobolev inequalities and the 1-biharmonic operator

Abstract

In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1. We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1-data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1, 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.

Keywords

• sobolev space embedding
• 1-biharmonic operator
• faber–krahn type inequality
• 46e35
• 35g15
• 35p30
• 49j52