Best constants in a borderline case of second-order Moser type inequalities
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 73-93.

We study optimal embeddings for the space of functions whose Laplacian Δu belongs to ${L}^{1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^{N}$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space ${W}^{2,1}\left(\Omega \right)$ 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 ${L}_{\mathrm{𝑒𝑥𝑝}}\left(\Omega \right)$. On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem $\Delta u=f\left(x\right)\in {L}^{1}\left(\Omega \right)$, $u=0$ 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.

DOI: 10.1016/j.anihpc.2009.07.006
Classification: 46E35,  35B65
Keywords: Sobolev embeddings, Pohožaev, Strichartz and Trudinger–Moser inequalities, Best constants, Elliptic equations, Regularity estimates in ${L}^{1}$, Brezis–Merle type results
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Cassani, Daniele; Ruf, Bernhard; Tarsi, Cristina. Best constants in a borderline case of second-order Moser type inequalities. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 1, pp. 73-93. doi : 10.1016/j.anihpc.2009.07.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.07.006/

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