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

Abstract

We study optimal embeddings for the space of functions whose Laplacian $\Delta u$ 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 $L_{\exp }(\Omega )$. 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 $\partial \Omega$; 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.