| Contents |
| We study optimal embeddings for the space of functions whose Laplacian 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 all second order
derivatives are taken into account. In particular, in the limiting Sobolev
case, when N = 2, we establish a sharp embedding inequality into the Zygmund
space $L_{\text{exp}}(\Omega)$. This result enables us to improve the Brezis-Merle regularity
estimate for the Dirichlet problem $\Delta u = f(x) \in L^1(\Omega), u = 0$ on $\partial \Omega$.
We then study the operator associated to related minimization problems, the {\it 1-biharmonic operator}
under various boundary conditions.
\par \bigskip \noindent
We consider the problem of finding the optimal constant for the embedding of the space
\[ W^{2,1}_\Delta(\Omega) := \left\{ u \in W^{1,1}_0(\Omega)\,|\,\Delta u \in L^1(\Omega)\right\} \]
into the space $L^1(\Omega)$, where $\Omega\subset \mathbb R^n$ is a
bounded convex domain, or a bounded domain with boundary of class
$C^{1,\alpha}$. This is equivalent to find the first eigenvalue of
the 1-biharmonic operator under (generalized) Navier boundary
conditions. In this paper we provide an interpretation for the
eigenvalue problem,
we show some properties of the first eigenfunction, we prove an inequality of Faber-Krahn type, and
we compute the first eigenvalue and the associated eigenfunction explicitly for a ball, and in terms of the torsion function for general domains.
\par \bigskip |
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