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Giuseppe Buttazzo

Publications and source records attributed to Giuseppe Buttazzo.

At least 55 records · Page 3Linked to original sources
arXiv2016

Optimal Control Problems in Transport Dynamics

In the present paper we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones.

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arXiv2016

Continuity and estimates for multimarginal optimal transportation problems with singular costs

We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.

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arXiv2016

Worst-case shape optimization for the Dirichlet energy

We consider the optimization problem for a shape cost functional $F(Ω,f)$ which depends on a domain $Ω$ varying in a suitable admissible class and on a "right-hand side" $f$. More precisely, the cost functional $F$ is given by an integral which involves the solution $u$ of an elliptic PDE in $Ω$ with right-hand side $f$; the boundary conditions considered are of the Dirichlet type. When the function $f$ is only known up to some degree of uncertainty, our goal is to obtain the existence of an optimal shape in the worst possible situation. Some numerical simulations are provided, showing the difference in the optimal shape between the case when $f$ is perfectly known and the case when only the worst situation is optimized.

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arXiv2016

Symmetry breaking for a problem in optimal insulation

We consider the problem of optimally insulating a given domain $Ω$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting $Ω$ to vary too.

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arXiv2015

A free boundary problem arising in PDE optimization

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form $$\sup_{\int_Dθ\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+θ}{2}|\nabla u|^2-fu\Big)\,dx.$$ We prove the existence of an optimal reinforcement $θ$ and that it has some higher integrability properties. We also provide some numerical computations for $θ$ and $u$.

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arXiv2015

Optimal design problems for Schrödinger operators with noncompact resolvents

We consider optimization problems for cost functionals which depend on the negative spectrum of Schrödinger operators of the form $-Δ+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.

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arXiv2014

Improved energy bounds for Schrödinger operators

Given a potential $V$ and the associated Schrödinger operator $-Δ+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ 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.

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arXiv2014

The spectral drop problem

We consider spectral optimization problems of the form $$\min\Big\{λ_1(Ω;D):\ Ω\subset D,\ |Ω|=1\Big\},$$ where $D$ is a given subset of the Euclidean space $\mathbb{R}^d$. Here $λ_1(Ω;D)$ is the first eigenvalue of the Laplace operator $-Δ$ with Dirichlet conditions on $\partialΩ\cap D$ and Neumann or Robin conditions on $\partialΩ\cap\partial D$. The equivalent variational formulation $$λ_1(Ω;D)=\min\left\{\int_Ω|\nabla u|^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{ on }\partialΩ\cap D,\ \|u\|_{L^2(Ω)}=1\right\}$$ reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.

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arXiv2014

Optimal regions for congested transport

We consider a given region $Ω$ where the traffic flows according to two regimes: in a region $C$ we have a low congestion, where in the remaining part $Ω\setminus C$ the congestion is higher. The two congestion functions $H_1$ and $H_2$ are given, but the region $C$ has to be determined in an optimal way in order to minimize the total transportation cost. Various penalization terms on $C$ are considered and some numerical computations are shown.

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arXiv2014

Stationary configurations for the average distance functional and related problems

For a functional defined on the class of closed one-dimensional connected subsets of ${\mathbb R}^n$ we consider the corresponding minimization problem and we give suitable first order necessary conditions of optimality. The cases studied here are the average distance functional arising in the mass transportation theory, and the energy related to an elliptic PDE.

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arXiv2013

Optimal spatial pricing strategies with transportation costs

We consider an optimization problem in a given region $Q$ where an agent has to decide the price $p(x)$ of a product for every $x\in Q$. The customers know the pricing pattern $p$ and may shop at any place $y$, paying the cost $p(y)$ and additionally a transportation cost $c(x,y)$ for a given transportation cost function $c$. We will study two models: the first one where the agent operates everywhere on $Q$ and a second one where the agent operates only in a subregion. For both models we discuss the mathematical framework and we obtain an existence result for a pricing strategy which maximizes the total profit of the agent. We also present some particular cases where more detailed computations can be made, as the case of concave costs, the case of quadratic cost, and the onedimensional case. Finally we discuss possible extensions and developments, as for instance the case of Nash equilibria when more agents operate on the same market.

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arXiv2013

A Mass Transportation Model for the Optimal Planning of an Urban Region

We propose a model to describe the optimal distributions of residents and services in a prescribed urban area. The cost functional takes into account the transportation costs (according to a Monge--Kantorovich-type criterion) and two additional terms which penalize concentration of residents and dispersion of services. The tools we use are the Monge--Kantorovich mass transportation theory and the theory of nonconvex functionals defined on measures.

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arXiv2013

Asymptotic optimal location of facilities in a competition between population and industries

We consider the problem of optimally locating a given number $k$ of points in ${\mathbb R}^n$ for an integral cost function which takes into account two measures $φ^+$ and $φ^-$. The points represent for example new industrial facilities that have to be located, the measure $φ^+$ representing in this case already existing industries that want to be close to the new ones, and $φ^-$ representing private citizens who want to stay far away. The asymptotic analysis as $k\to\infty$ is performed, providing the asymptotic density of optimal locations.

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arXiv2013

The Monge-Kantorovich problem for distributions and applications

We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace $X(Ω)$ of first order distribution. A particular subclass $X_0^\sharp(Ω)$ of such distributions will be considered which includes the infinite sums of dipoles $\sum_k(δ_{p_k}-δ_{n_k})$ studied in \cite{P1, P2}. In spite of this weakened regularity, it is shown that an optimal transport density still exists among nonnegative finite measures. Some geometric properties of the Banach spaces $X(Ω)$ and $X_0^\sharp(Ω)$ can be then deduced.

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arXiv2013

Shape optimization problems on metric measure spaces

We consider shape optimization problems of the form $$\min\big\{J(Ω)\ :\ Ω\subset X,\ m(Ω)\le c\big\},$$ where $X$ is a metric measure space and $J$ is a suitable shape functional. We adapt the notions of $γ$-convergence and weak $γ$-convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.

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arXiv2013

Shape Optimization Problems for Metric Graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(Γ)\ :\ Γ\in{\mathcal A},\ {\mathcal H}^1(Γ)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(Γ)$ is the Dirichlet energy of $Γ$ defined through the Sobolev functions on $Γ$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

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arXiv2013

Spectral optimization problems for potentials and measures

In the present paper we consider spectral optimization problems involving the Schrödinger operator $-Δ+μ$ on $\R^d$, the prototype being the minimization of the $k$ the eigenvalue $λ_k(μ)$. Here $μ$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\ds \int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in $\R^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.

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