On the characterization of the compact embedding of Sobolev spaces

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of $p$-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

Overdetermined boundary value problems for the ∞\infty-Laplacian

We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $\Omega$ of $\R^n$ and discuss what kind of implications on the geometry of $\Omega$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.Comment: 9 pages, 1 figur

Asymptotics of an optimal compliance-location problem

We consider the problem of placing n small balls of given radius in a certain domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look at the asymptotics of the minimization problem, after properly scaling the functionals involved, and to the limit distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.Comment: 20 pages with 2 figures; final accepted version (minor changes, some extra details on the positivity assumption on $f$

Long-term planning versus short-term planning in the asymptotical location problem

Given the probability measure $\nu$ over the given region $\Omega\subset \R^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points \Om in order to minimize the average distance \Sigma\mapsto \int_\Om \dist(x,\Sigma) d\nu (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as $n\to\infty$, although the optimization costs in both cases have the same asymptotic orders of vanishing.Comment: for more pictures and some movies as well, see http://www.sissa.it/~brancoli

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

Elastic DVS Management in Processors with Discrete Voltage/Frequency Modes

Applying classical dynamic voltage scaling (DVS) techniques to real-time systems running on processors with discrete voltage/frequency modes causes a waste of computational resources. In fact, whenever the ideal speed level computed by the DVS algorithm is not available in the system, to guarantee the feasibility of the task set, the processor speed must be set to the nearest level greater than the optimal one, thus underutilizing the system. Whenever the task set allows a certain degree of flexibility in specifying timing constraints, rate adaptation techniques can be adopted to balance performance (which is a function of task rates) versus energy consumption (which is a function of the processor speed). In this paper, we propose a new method that combines discrete DVS management with elastic scheduling to fully exploit the available computational resources. Depending on the application requirements, the algorithm can be set to improve performance or reduce energy consumption, so enhancing the flexibility of the system. A reclaiming mechanism is also used to take advantage of early completions. To make the proposed approach usable in real-world applications, the task model is enhanced to consider some of the real CPU characteristics, such as discrete voltage/frequency levels, switching overhead, task execution times nonlinear with the frequency, and tasks with different power consumption. Implementation issues and experimental results for the proposed algorithm are also discussed