Abstract
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.
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