1,854 research outputs found
Spectral Optimization Problems
In this survey paper we present a class of shape optimization problems where
the cost function involves the solution of a PDE of elliptic type in the
unknown domain. In particular, we consider cost functions which depend on the
spectrum of an elliptic operator and we focus on the existence of an optimal
domain. The known results are presented as well as a list of still open
problems. Related fields as optimal partition problems, evolution flows,
Cheeger-type problems, are also considered.Comment: 42 pages with 8 figure
On the characterization of the compact embedding of Sobolev spaces
For every positive regular Borel measure, possibly infinite valued, vanishing
on all sets of -capacity zero, we characterize the compactness of the
embedding W^{1,p}({\bf R}^N)\cap L^p ({\bf R}^N,\mu)\hr L^q({\bf R}^N) in
terms of the qualitative behavior of some characteristic PDE. This question is
related to the well posedness of a class of geometric inequalities involving
the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced
by Polya and Szeg\"o in 1951. In particular, we prove that finite torsional
rigidity of an arbitrary domain (possibly with infinite measure), implies the
compactness of the resolvent of the Laplacian.Comment: 19 page
Shape optimization problems on metric measure spaces
We consider shape optimization problems of the form where is a metric measure space
and is a suitable shape functional. We adapt the notions of
-convergence and weak -convergence to this new general abstract
setting to prove the existence of an optimal domain. Several examples are
pointed out and discussed.Comment: 27 pages, the final publication is available at
http://www.journals.elsevier.com/journal-of-functional-analysis
Improved energy bounds for Schr\"odinger operators
Given a potential and the associated Schr\"odinger operator ,
we consider the problem of providing sharp upper and lower bound on the energy
of the operator. It is known that if for example or enjoys
suitable summability properties, the problem has a positive answer. In this
paper we show that the corresponding isoperimetric-like inequalities can be
improved by means of quantitative stability estimates.Comment: 31 page
Overdetermined boundary value problems for the -Laplacian
We consider overdetermined boundary value problems for the -Laplacian
in a domain of and discuss what kind of implications on the
geometry of the existence of a solution may have. The classical
-Laplacian, the normalized or game-theoretic -Laplacian and the
limit of the -Laplacian as are considered and provide
different answers.Comment: 9 pages, 1 figur
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