## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
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DCDS

We study the properties of the transport density measure
in the Monge-Kantorovich
optimal mass transport problem in the presence of so-called Dirichlet
constraint, i.e. when some
closed set is given
along which the cost of transportation is zero. The Hausdorff dimension
estimates, as well as summability and higher regularity properties
of the transport density are studied. The uniqueness of the
transport density is proven in the case when the masses to be transported
are represented by measures absolutely continuous with respect
to the Lebesgue measure.

DCDS-B

Radiotherapy is an important clinical tool to fight malignancies. To do so, a key point consists in selecting a suitable radiation dose that could achieve tumour control without inducing significant damage to surrounding healthy tissues. In spite of recent significant advances, any radiotherapy planning in use relies principally on experience-based decisions made by clinicians among several possible choices.

In this work we consider a mathematical problem related to that decision-making process. More precisely, we assume that a well-defined target region, called planning target volume (PTV), is given. We then consider the question of determining which radiation distribution is able to achieve a maximum impact on tumour cells and a minimum one in healthy ones. Such dose distribution is defined as the solution of a multi-parameter minimization problem over the PTV and healthy tissues, subject to a number of constraints arising from clinical and technical requirements. For any choice of parameters, sufficient conditions for the existence of a unique solution of that problem are derived. Such solution is then approximated by means of a suitable numerical algorithm. Finally, some examples are considered, on which the dependence on model parameters of different clinical efficiency indexes is discussed.

In this work we consider a mathematical problem related to that decision-making process. More precisely, we assume that a well-defined target region, called planning target volume (PTV), is given. We then consider the question of determining which radiation distribution is able to achieve a maximum impact on tumour cells and a minimum one in healthy ones. Such dose distribution is defined as the solution of a multi-parameter minimization problem over the PTV and healthy tissues, subject to a number of constraints arising from clinical and technical requirements. For any choice of parameters, sufficient conditions for the existence of a unique solution of that problem are derived. Such solution is then approximated by means of a suitable numerical algorithm. Finally, some examples are considered, on which the dependence on model parameters of different clinical efficiency indexes is discussed.

DCDS

In this paper we analyze the relaxed form of a shape optimization
problem with state equation
$$
\begin{equation}
\left\{\begin{array}{ll}
-div\big(a(x)Du\big)=f\qquad\hbox{in }D\\
\hbox{boundary conditions on }\partial D.
\end{array}
\right.
\end{equation}
$$
The new fact is that the term $f$ is only known up to a random
perturbation $\xi(x,\omega)$. The goal is to find an optimal
coefficient $a(x)$, fulfilling the usual constraints $\alpha\le
a\le\beta$ and $\displaystyle\int_D a(x)\,dx\le m$, which
minimizes a cost function of the form
$$\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big)\,dx\,dP(\omega).$$
Some numerical examples are shown in the last section, which
evidence the previous analytical results.

NHM

We consider the problem of the optimal location of a Dirichlet region in a two-dimensional domain $\Omega$ subject to a force $f$ in order to minimize the compliance of the configuration. The class of admissible Dirichlet regions among which we look for the optimum consists of all one-dimensional connected sets (networks) of a given length $L$. Then we let $L$ tend to infinity and look for the $\Gamma$-limit of suitably rescaled functionals, in order to identify the asymptotical distribution of the optimal networks. The asymptotically optimal shapes are discussed as well and links with average distance problems are provided.

DCDS

We study the variational
convergence, as $\h \rightarrow \infty$, of
a sequence of optimal control problems $(\mathcal{P}_h)$ with abstract
state equations $A_h(y)=B_h(u)$, where $A_h$ are $G$-converging and
the operators $B_h$ acting on the
controls are supposed continuously
converging, or nonlinear but local, or linear but possibly nonlocal.

DCDS

To every distance $d$ on a given open set $\Omega\subseteq\mathbb R^n$,
we may associate several kinds of variational problems. We show
that, on the class of all geodesic distances $d$ on $\Omega$ which
are bounded from above and from below by fixed multiples of the
Euclidean one, the uniform convergence on compact sets turns out
to be equivalent to the $\Gamma$-convergence of each of the
corresponding variational problems under consideration.

DCDS

In the paper a model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra

*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.
keywords:
routing cost
,
airfreight systems.
,
Gamma-convergence
,
Location problems
,
transport problems

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