Giuseppe Buttazzo | University of Pisa (original) (raw)

Papers by Giuseppe Buttazzo

arXiv (Cornell University), Dec 15, 2010

In this survey paper we present a class of shape optimization problems where the cost function in... more 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.

arXiv (Cornell University), Sep 13, 2019

We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the line... more 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 R d , which implies a lack of compactness in several crucial points. In the last section we present some numerical simulations.

arXiv (Cornell University), Apr 16, 2013

We present some new problems in spectral optimization. The first one consists in determining the ... more 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 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.

arXiv (Cornell University), Jan 13, 2020

We investigate extremal properties of shape functionals which are products of Newtonian capacity ... more We investigate extremal properties of shape functionals which are products of Newtonian capacity cap (Ω), and powers of the torsional rigidity T (Ω), for an open set Ω ⊂ R d with compact closure Ω, and prescribed Lebesgue measure. It is shown that if Ω is convex then cap (Ω)T q (Ω) is (i) bounded from above if and only if q ≥ 1, and (ii) bounded from below and away from 0 if and only if q ≤ 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 < d−2 2(d−1). If q ≤ 0, then the product is minimised among all bounded sets by a ball of measure 1.

In this talk we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by... more In this talk we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a connected one-dimensional structure. 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 structures when their total length becomes large.

The Mathematical Intelligencer, Sep 1, 1993

In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible med... more In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible medium. In his words (from his Principia Mathematica): If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameter move with equal velocities in the direction of the axis of the cylinder, (then) the resistance of the globe will be half as great as that of the cylinder.... I reckon that this proposition will be not without application in the building of ships.

HAL (Le Centre pour la Communication Scientifique Directe), Jan 31, 2023

Identifying Blaschke-Santaló diagrams is an important topic that essentially consists in determin... more Identifying Blaschke-Santaló diagrams is an important topic that essentially consists in determining the image Y = F (X) of a map F : X → R d , where the dimension of the source space X is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, X can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number N of points x 1 ,. .. , x N in X and plotting them in the target space R d , produces in many cases areas in Y of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points x i in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set Y by a relatively small number N of samples which is very useful when the dimension of the source space X is large (or even infinite) and the evaluation of F (x i) is costly. Our method consists in a suitable use of Centroidal Voronoi Tessellations which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.

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arXiv (Cornell University), Nov 13, 2009

We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of R n and d... more We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of R n and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian and the limit of the p-Laplacian as p → ∞ are considered and provide different answers, even if we restrict our domains to those that have only web-functions as solutions.

arXiv (Cornell University), Dec 10, 2022

We consider the shape optimization problems for the quantities λ(Ω)T q (Ω), where Ω varies among ... more We consider the shape optimization problems for the quantities λ(Ω)T q (Ω), where Ω varies among open sets of R d with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case q > 1. We prove that for q large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among nearly spherical domains.

arXiv (Cornell University), Feb 5, 2023

In this paper we consider optimal control problems where the control variable is a potential and ... more In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of a Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.

arXiv (Cornell University), Dec 18, 2022

We prove the first regularity theorem for the free boundary of solutions to shape optimization pr... more We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain Ω is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on Ω. The main feature of these functionals is that the minimality of a domain Ω cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions j(u, x) = −g(x)u + Q(x), where u is the solution of the PDE −∆u = f with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of Ω and then we use the stability of Ω with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the onephase Bernoulli problem and according to the blow-up limits, we decompose ∂Ω into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of ∂Ω we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove C ∞ regularity of the regular part of the free boundary when the data are smooth. Contents 1. Introduction 2 2. First and second variations under inner perturbations 10 3. Lipschitz regularity and non-degeneracy of the state functions 18 4. Compactness and convergence of blow-up sequences 21 5. Regular and singular parts of the free boundary 27 6. Regularity of Reg(∂Ω) 29 7. Stable homogeneous solutions of the one-phase Bernoulli problem 31 8. Hausdorff dimension of Sing(∂Ω) 40 9. Existence of optimal sets in R d and proof of Theorem 1.6 43 Appendix A. An optimization problem in heat conduction 47 References 50

arXiv (Cornell University), Nov 9, 2022

Starting from the Brock's construction of Continuous Steiner Symmetrization of sets, the problem ... more 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.

arXiv (Cornell University), Jul 28, 2022

We consider optimal transport problems where the cost for transporting a given probability measur... more We consider optimal transport problems where the cost for transporting a given probability measure µ 0 to another one µ 1 consists of two parts: the first one measures the transportation from µ 0 to an intermediate (pivot) measure µ to be determined (and subject to various constraints), and the second one measures the transportation from µ to µ 1. This leads to Wasserstein interpolation problems under constraints for which we establish various properties of the optimal pivot measures µ. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Wasserstein constrained interpolation problems.

Analisi matematica.-The bounce problem, on n-dimensional Riemannian manifolds. Nota di G i u s e ... more Analisi matematica.-The bounce problem, on n-dimensional Riemannian manifolds. Nota di G i u s e p p e B u t t a z z o e D a n i l o P e rc iv a l l e , presentata W dal Corrips. E. D e G io r g i. R ia ssu n to .-In questo lavoro vengono generalizzati i risultati relativi al problema del rimbalzo unidimensionale studiato in [5]. Precisamente si considera un punto mobile su una varietà Riemanniana V w-dimensionale, soggetto all'azione di un potenziale varia bile nel tempo e vincolato a restare in una parte W di V avente un bordo di classe C3 contro cui il punto « rimbalza ». Lo studio del problema richiede l'uso di metodi di T-convergenza del tipo usato in [5], metodi che sembrano caratteristici per lo studio di problemi in cui può mancare l'unicità della soluzione o la sua dipendenza continua dai dati.

ESAIM: Mathematical Modelling and Numerical Analysis, 1997

Un problème d'optimisation de plaques RAIRO-Modélisation mathématique et analyse numérique, tome ... more Un problème d'optimisation de plaques RAIRO-Modélisation mathématique et analyse numérique, tome 31, n o 2 (1997), p. 167-184. <http://www.numdam.org/item?id=M2AN_1997__31_2_167_0> © AFCET, 1997, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATHEMATICAL MODELUNG AND NUMER1CAL ANALYSIS MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

Springer Optimization and Its Applications, 2012

The Dirichlet problem for the bi-harmonic equation is considered as the Kirchhoff model of an iso... more The Dirichlet problem for the bi-harmonic equation is considered as the Kirchhoff model of an isotropic elastic plate clamped at its edge. The plate is supported at certain points P 1,…,P J , that is, the deflexion u(x) satisfies the Sobolev point conditions u(P 1)=⋯=u(P J )=0. The optimal location of the support points is discussed such that either the compliance functional or the minimal deflexion functional attains its minimum.

1. An Appreciation of James Serrin.- 2. On Keplerian N-Body Type Problems.- 3. Invariance and Bal... more 1. An Appreciation of James Serrin.- 2. On Keplerian N-Body Type Problems.- 3. Invariance and Balance in Continuum Mechanics.- 4. Entropy Numbers, Approximation Numbers, and Embeddings.- 5. Some Regularity Properties of Locally Weakly Invertible Maps.- 6. Some Recent Results on Saint-Venant's Principle.- 7. Some Results on Modifications of Three-Dimensional Navier-Stokes Equations.- 8. An Integral Equation in Probability.- 9. Self-Similar Solutions of the Second Kind.- 10. On the Problem of a Moving Contact Angle.- 11. Space, Time, and Energy: Lectio Doctoralis.