Abstract:
We consider a class of one--dimensional non--convex non--coercive problems in the Calculus of Variations. We prove an existence result for this class of problems using a Liapunov type theorem on the range of non--atomic measures.

Abstract:
Segregation of chromosomes to the two opposite poles of a dividing cell is central to cellular reproduction. Fittingly, the mitotic spindle, which is primarily responsible for progressive separation of sister-chromatids during anaphase, is a tension-ridden, bipolar structure with centrosomes constituting the two poles. Centrosomes are also the microtubule organizing centers (MTOC) of a spindle [1]. In a typical metaphase spindle, the two centrosomes are in a 'face-to-face' configuration, separated by a set of overlapping microtubules, emanating from each centrosome towards the other (pole-to-pole microtubules) [2]. A second set of microtubules, called astral microtubules, radiate from each centrosome towards the cell cortex. A third set, termed kinetochore microtubules, emanate from the centrosomes and eventually attach to duplicated chromosomes at the kinetochores such that each sister kinetochore is linked to one pole that is directly facing it while the other is attached to the opposite pole. Due to the dynamic nature of the microtubules and the activities of the proteins associated with them (microtubule associated proteins or MAPS), the spindle poles (centrosomes) tend to move away from each other, causing the sister chromatids to be pulled in the opposite direction. In a metaphase spindle, this tendency is opposed by proteins known as cohesins that tether the sister chromatids together. Such opposing forces within the spindle are what make it a tension-ridden structure.The centrosome is pivotal to the biogenesis of the mitotic spindle. In many animals, assembly of the first spindle in the fertilized egg is dependent on the MTOC (in the form of a basal body centriole) contributed by the sperm since oocyte centrosome degenerates sometime during oogenesis. Hence, during fertilization, the sperm contributes not only DNA but also MTOC for construction of a spindle [3]. The incoming centriole then recruits maternal components that constitute the pericentriolar mater

Abstract:
Many Pigmented lesions of nonmelanocytic origin can mimic clinically melanocytic lesions including malignant melanoma. A histological interpretation by pathology is helpful in the diagnosis and management of these lesions. The cases during a two year period from January 1999 to December 2000 were reviewed to assess the prevalence of lesions with pigmented variants where histopathological examination helped to confirm/refute the clinical diagnosis. The most common lesion presented with such diagnostic difficulty clinically was seborrhoeic keratosis. Other lesions observed in the study included basal cell carcinoma (12), actinic keratosis (3) and dermatofibrosarcoma protuberans (3). The total number of cases studied was 26. Adherence to strict diagnostic criteria helped towards the correct diagnosis. As 50% of the lesions had pigmentation and 30% had a clinical diagnosis of melanoma, histopathologic evolution was crucial to avoid overdiagnosis of melanoma and to provide reassurance in benign lesions.

Abstract:
We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $\Omega\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $\Omega$. In the special case when $\Omega$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.

Abstract:
We prove that, if $\Omega\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial \Omega$ of the function $$\varphi(y) = \int_0^{\lambda(y)} \prod_{j=1}^{n-1}[1-t \kappa_j(y)]\, dt$$ implies that $\Omega$ is a ball. Here $k_j(y)$ and $\lambda(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \in \partial \Omega$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \to + \infty$ of Serrin's symmetry problem for the $p$-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain.

Abstract:
We consider a system of PDEs of Monge-Kantorovich type that, in the isotropic case, describes the stationary configurations of two-layers models in granular matter theory with a general source and a general boundary data. We propose a new weak formulation which is consistent with the physical model and permits us to prove existence and uniqueness results.

Abstract:
Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $\Omega \subset \mathbb{R}^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$ with constant source term admits a viscosity solution depending only on the distance from $\partial \Omega$. This problem was previously addressed and studied by Buttazzo and Kawohl. In the light of some geometrical achievements reached in our recent paper "On the characterization of some classes of proximally smooth sets", we revisit the results obtained by Buttazzo and Kawohl and we prove strengthened versions of them, where any regularity assumption on the domain and on the solution is removed. Our results require a delicate analysis based on viscosity methods. In particular, we need to build suitable viscosity test functions, whose construction involves a new estimate of the distance function near singular points.

Abstract:
We give a complete characterization, as "stadium-like domains", of convex subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $\Omega$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet problem is of class $C^{1,1} (\Omega)$.

Abstract:
Let $F:[0,T]\times\R^n\mapsto 2^{\R^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if $F$ satisfies the following Lipschitz Selection Property: \begin{itemize} \item[{(LSP)}] {\sl For every $t,x$, every $y\in \overline{co} F(t,x)$ and $\varepsilon>0$, there exists a Lipschitz selection $\phi$ of $\overline{co}F$, defined on a neighborhood of $(t,x)$, with $|\phi(t,x)-y|<\varepsilon$.} \end{itemize} then there exists a measurable selection $f$ of $ext F$\ such that, for every $x_0$, the Cauchy problem $$ \dot x(t)=f(t,x(t)),\qquad\qquad x(0)=x_0 $$ has a unique Caratheodory solution, depending continuously on $x_0$. We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class, for which (LSP) holds, consists of those continuous multifunctions $F$ whose values are compact and have convex closure with nonempty interior.

Abstract:
We consider the Bean's critical state model for anisotropic superconductors. A variational problem solved by the quasi--static evolution of the internal magnetic field is obtained as the $\Gamma$-limit of functionals arising from the Maxwell's equations combined with a power law for the dissipation. Moreover, the quasi--static approximation of the internal electric field is recovered, using a first order necessary condition. If the sample is a long cylinder subjected to an axial uniform external field, the macroscopic electrodynamics is explicitly determined.