arXiv subjects
Publications and source records attributed to Giuseppe Buttazzo.
Starting from the Brock's construction of Continuous Steiner Symmetrization of sets, the problem of modifying continuously a given domain up to obtain a ball, preserving its measure and with decreasing first eigenvalue of the Laplace operator, is considered. For a large class of cases it is shown this is possible, while the general question remains still open.
We study a general version of the Cheeger inequality by considering the shape functional $\mathcal{F}_{p,q}(Ω)=λ_p^{1/p}(Ω)/λ_q(Ω)^{1/q}$. The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of all domains $Ω$ of $\mathbb{R}^d$ and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for $\mathcal{F}_{p,q}$ is discussed.
We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $p\le d$, where additional assumptions have to be made on the data.
In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in \cite{bro95,bro00}. It transforms every domain $Ω\subset\subset\mathbb{R}^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsion respectively decrease and increase. While this does not provide, in general, a $γ$-continuous map $t\mapstoΩ_t$, it can be slightly modified so to obtain the $γ$-continuity for a $γ$-dense class of domains $Ω$, namely, the class of polyedral sets in $\mathbb{R}^d$. This allows to obtain a sharp characterization of the Blaschke-Santaló diagram of torsion and eigenvalue.
We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.
A shape optimization problem arising from the optimal reinforcement of a membrane by means of one-dimensional stiffeners or from the fastest cooling of a two-dimensional object by means of ``conducting wires'' is considered. The criterion we consider is the maximization of the first eigenvalue and the admissible classes of choices are the one of one-dimensional sets with prescribed total length, or the one where the constraint of being connected (or with an a priori bounded number of connected components) is added. The corresponding relaxed problems and the related existence results are described.
We consider the problem of minimising or maximising the quantity $λ(Ø)T^q(Ø)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $λ(Ø)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(Ø)$, while $T(Ø)$ is the torsional rigidity of $Ø$. The optimisation problem above is considered in the class of {\it all domains} $Ø$, in the class of {\it convex domains} $Ø$, and in the class of {\it thin domains}. The full Blaschke-Santaló diagram for $λ(Ø)$ and $T(Ø)$ is obtained in dimension one, while for higher dimensions we provide some bounds.
We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the linear part of order zero given by a potential V . The main goal is to study the optimization problem for an integral cost depending on the solution uV , when V varies in a suitable class of admissible potentials. These problems can be seen as the natural extension of shape optimization problems to the framework of potentials. The main result is an existence theorem for optimal potentials, and the main difficulty is to work in the whole Euclidean space Rd, which implies a lack of compactness in several crucial points. In the last section we present some numerical simulations.
We propose a duality theory for multi-marginal repulsive cost that appear in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the $N$-marginals relaxed cost in terms of a stratification formula which takes into account all $k$ interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.
In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected onedimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when their total length becomes large.
In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.
We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $Ω$ that varies over all subdomains of a given bounded domain $D$ of ${\bf R}^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.
We consider optimal control problems where the state equation is an elliptic PDE of a Schrödinger type, governed by the Laplace operator $-Δ$ with the addition of a potential V, and the control is the potential V itself, that may vary in a suitable admissible class. In a previous paper (Ref. [7]) an existence result was established under a monotonicity assumption on the cost functional, which occurs if the data do not change sign. In the present paper this sign assumption is removed and the existence of an optimal potential is still valid. Several numerical simulations, made by FreeFem++, are shown
We consider two optimization problems in thermal insulation: in both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The total mass of the insulating material is prescribed.. The first problem deals with the case in which a given heat source is present, while in the second one there are no heat sources and the goal is to have the slowest decay of the temperature. In both cases an optimal distribution of the insulator around the thermal body exists; when the body has a circular symmetry, in the first case a constant heat source gives a constant thickness as the optimal solution, while surprisingly this is not the case in the second problem, where the circular symmetry of the optimal insulating layer depends on the total quantity of insulator at our disposal. A symmetry breaking occurs when this total quantity is below a certain threshold. Some numerical computations are also provided, together with a list of open questions.
The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry breaking occurs for the case of small available insulation masses and provide insight in the geometry of optimal films. An experimental shape optimization indicates that convex bodies with one axis of symmetry have favorable insulation properties.
In this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal domain exists. We also deduce some necessary conditions of optimality for the optimal domain. The results are applied to show the existence of an optimal domain in the case where the cost functional is completely identified, while the right-hand side in the state equation is only known up to a probability P in the space $L^2$.
In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic non linear operators.