Best constants in a borderline case of second-order Moser type inequalities

Cassani, Daniele ; Ruf, Bernhard ; Tarsi, Cristina

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 73-93 / Harvested from Numdam

Résumé

We study optimal embeddings for the space of functions whose Laplacian Δu belongs to $ {L}^{1}(\Omega )$, where $ \Omega \subset {\mathbb{R}}^{N}$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space $ {W}^{2,1}(\Omega )$ in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when $ N=2$, we establish a sharp embedding inequality into the Zygmund space $ {L}_{\mathit{exp}}(\Omega )$. On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem $ \mathrm{\Delta }u=f(x)\in {L}^{1}(\Omega )$, $ u=0$ on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension $ N⩾3$ are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.

Détail

Publié le : 2010-01-01

Zbl 1194.46048 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2009.07.006
Classification:  46E35,  35B65

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