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Giuseppe Buttazzo

Publications and source records attributed to Giuseppe Buttazzo.

69 records · Page 4Linked to original sources
arXiv2013

Optimal location problems with routing cost

In the paper a model problem for the location of a given number $N$ of points in a given region $Ω$ and with a given resources density $ρ(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra {\it routing cost} is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown.

math.OC↗
arXiv2013

Optimal Potentials For Schrodinger Operators

We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

math.AP↗
arXiv2013

Stationary Configuration of some Optimal Shaping

We consider the problem of optimal location of a Dirichlet region in a $d$-dimensional domain $Ω$ subjected to a given right-hand side $f$ in order to minimize some given functional of the configuration. While in the literature the Dirichlet region is usually taken $d-1$ dimensional, in this shape optimization problems, we consider two classes of control variables, namely the class of one dimensional closed connected sets of finite one dimensional Hausdorff measure and the class of sets of points of finite cardinality, and we give a necessary condition of optimality.

math.OC↗
arXiv2013

Some New Problems in Spectral Optimization

We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the {\it metric Laplacian}, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carathéodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schrödinger potential in suitable classes.

math.OC↗
arXiv2012

Optimal-transport formulation of electronic density-functional theory

The most challenging scenario for Kohn-Sham density functional theory, that is when the electrons move relatively slowly trying to avoid each other as much as possible because of their repulsion (strong-interaction limit), is reformulated here as an optimal transport (or mass transportation theory) problem, a well established field of mathematics and economics. In practice, we show that solving the problem of finding the minimum possible internal repulsion energy for $N$ electrons in a given density $ρ(\rv)$ is equivalent to find the optimal way of transporting $N-1$ times the density $ρ$ into itself, with cost function given by the Coulomb repulsion. We use this link to put the strong-interaction limit of density functional theory on firm grounds and to discuss the potential practical aspects of this reformulation.

cond-mat.str-el↗
arXiv2012

Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(ρ,v)$, depending on the density $ρ$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.

math.OC↗
arXiv2011

Shape flows for spectral optimization problems

We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the $γ$-convergence dissipation distance and with the evolution of domains in spectral optimization problems.

math.AP↗
arXiv2011

Shape Optimization Problems with Internal Constraint

We consider shape optimization problems with internal inclusion constraints, of the form $$\min\big\{J(Ω)\ :\ \Dr\subsetΩ\subset\R^d,\ |Ω|=m\big\},$$ where the set $\Dr$ is fixed, possibly unbounded, and $J$ depends on $Ω$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

math.AP↗
arXiv2010

Weak Optimal Controls in Coefficients for Linear Elliptic Problems

In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in $L^1(Ω)$. Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.

math.OC↗
arXiv2010

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.

math.OC↗
arXiv2010

A Benamou-Brenier approach to branched transport

The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length

math.OC↗
arXiv2010

Optimal Shape for Elliptic Problems with Random Perturbations

In this paper we analyze the relaxed form of a shape optimization problem with state equation $\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.$ The new fact is that the term $f$ is only known up to a random perturbation $ξ(x,ω)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $α\le a\leβ$ and $\displaystyle\int_D a(x) dx\le m$, which minimizes a cost function of the form $$\int_Ω\int_Dj\big(x,ω,u_a(x,ω)\big) dx dP(ω).$$ Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.

math.OC↗
arXiv2009

On some rescaled shape optimization problems

We consider Cheeger-like shape optimization problems of the form $$\min\big\{|Ω|^αJ(Ω) : Ω\subset D\big\}$$ where $D$ is a given bounded domain and $α$ is above the natural scaling. We show the existence of a solution and analyze as $J(Ω)$ the particular cases of the compliance functional $C(Ω)$ and of the first eigenvalue $λ_1(Ω)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.

math.OC↗
arXiv2009

Minimization of $λ_2(Ω)$ with a perimeter constraint

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $γ$ lower semicontinuous.

math.AP↗