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.

Limiting Sobolev inequalities and the 1-biharmonic operator / E. Parini, B. Ruf, C. Tarsi. - In: ADVANCES IN NONLINEAR ANALYSIS. - ISSN 2191-9496. - 3:S1(2014 Sep), pp. 19-36.

Limiting Sobolev inequalities and the 1-biharmonic operator

B. Ruf;C. Tarsi
2014

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.
Sobolev space embedding; 1-biharmonic operator; Faber-Krahn type inequality; clamped plate; isoperimetric-inequalities; rayleighs conjecture; Boundary-conditions; elliptic-equations; 1-Laplace operator; Eigenvalue problem; torsion function; dimensions; laplacian
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/251043
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