Some remarks on the optimal design of periodically reinforced structures
Local Mappings on spaces of differentiable functions.
Errata to: Local Mappings on Spaces of Differentiable Functions [1].
Optimal networks for mass transportation problems
In the framework of transport theory, we are interested in the following optimization problem: given the distributions of working people and of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of from with respect to a metric which depends on the transportation network....
Relaxation for a class of nonconvex functionals defined on measures
Reinforcement problems in the calculus of variations
Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem
In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried...
The bounce problem, on n-dimensional Riemannian manifolds
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 -dimensionale, soggetto all’azione di un potenziale variabile nel tempo e vincolato a restare in una parte di avente un bordo di classe contro cui il punto «rimbalza». Lo studio del problema richiede l’uso di metodi di -convergenza del tipo usato in [5], metodi che sembrano caratteristici...
Quasilinear elliptic equations with discontinuous coefficients
We prove an existence result for equations of the form where the coefficients satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable . Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients are supposed only Borel functions
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 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.
Quasilinear elliptic equations with discontinuous coefficients
We prove an existence result for equations of the form where the coefficients satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable . Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients are supposed only Borel functions
The bounce problem, on n-dimensional Riemannian manifolds
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 -dimensionale, soggetto all’azione di un potenziale variabile nel tempo e vincolato a restare in una parte di avente un bordo di classe contro cui il punto «rimbalza». Lo studio del problema richiede l’uso di metodi di -convergenza del tipo usato in [5], metodi che sembrano caratteristici...
Characterization of optimal shapes and masses through Monge-Kantorovich equation
We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.
Optimal networks for mass transportation problems
In the framework of transport theory, we are interested in the following optimization problem: given the distributions µ of working people and µ of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of µ from µ with respect to a metric which depends on the transportation network. ...
Shape optimization problems for metric graphs
): ∈ 𝒜, ℋ() = }, where ℋ ,, } ⊂ R . The cost functional ℰ() is the Dirichlet energy of defined through the Sobolev functions on vanishing on the points . 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.
Stationary configurations for the average distance functional and related problems
Mean curvature of a measure and related variational problems
Singular perturbation problems in the calculus of variations
Semicontinuity in for polyconvex integrals
Viene studiata la semicontinuità rispetto alla topologia di per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.
Completion by Gamma-convergence for optimal control problems
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