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) 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.

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