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Article Dans Une Revue Advances in Nonlinear Analysis Année : 2014

Limiting Sobolev inequalities and the 1-biharmonic operator

Enea Parini
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Bernhard Ruf
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Cristina Tarsi
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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.

Dates et versions

hal-03975914 , version 1 (06-02-2023)

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Enea Parini, Bernhard Ruf, Cristina Tarsi. Limiting Sobolev inequalities and the 1-biharmonic operator. Advances in Nonlinear Analysis, 2014, 3 (S1), pp.s19-s36. ⟨10.1515/anona-2014-0007⟩. ⟨hal-03975914⟩

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