Lorenzo Brasco: Torsional rigidity and embeddings for homogeneous Sobolev spaces

Let $\Omega$ be a generic open set of the Euclidean space. We consider the
{\it homogeneous Sobolev space}, defined by the completion of the space of
compactly supported smooth functions with respect to the $L^p$ norm of the
gradient.

We give a characterization of the continuous (or compact) embedding of this
space into $L^q(\Omega)$, in terms of the summability of the so-called {\it
$p-$torsion function} of $\Omega$. We also introduce a new Hardy-type
inequality, which plays an important role in the proofs.

Finally, in the Hilbertian case we give some applications of our Hardy-type
inequality to ground state estimates for Schr\"odinger operators.

The results presented are contaied in a joint work with Berardo Ruffini
(Montpellier).