Limiting Sobolev inequalities and the 1-biharmonic operator

作者：Parini Enea^*; Ruf Bernhard; Tarsi Cristina

来源：Advances in Nonlinear Analysis, 2014, 3(0): S19-S36.

DOI：10.1515/anona-2014-0007

• Sobolev space embedding
• 1-biharmonic operator
• Faber-Krahn type inequality

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.

• 出版日期2014-9