Showing papers in "Ricerche Di Matematica in 2006"

Models of phase segregation and diffusion of atomic species on a lattice

Abstract: Two mathematical models for phase segregation and diffusion of an order parameter are derived, within one and the same continuum mechanical framework. These models are, respectively, of the Allen-Cahn type and of the Cahn-Hilliard type. Our framework is similar to that used in [1], in that a postulated balance of microforces plays a central role in both deductive paths, but differs from it, mainly in three ways: imbalance of entropy replaces for a dissipation inequality, whose form depends on the case, restricting the growth of free energy; balance of energy replaces for the mass balance introduced in [1] to arrive at (a generalization of) the C-H equation; and chemical potential is given the same role played by coldness in the deduction of the heat equation. When appropriate constitutive prescriptions are made, different in the cases of segregation and diffusion but consistent with the entropy imbalance, it is found that standard A-C and C-H processes are solutions of constant chemical potential of the corresponding generalized equations; in particular, the stationary solutions are the same. Keywords: Phase segregation, Diffusion, Allen-Cahn equation, Cahn-Hilliard equation, Phase-field methods Mathematics Subject Classification (2000): 74N25, 74A50, 35K60

Existence of static solutions of the semilinear Maxwell equations

Abstract: In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$ and the magnetic field ${\bf H}$ does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ $ abla\times $ ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$ is the gauge potential related to the magnetic field ${\bf H}$ ( ${\bf H}= abla\times {\bf A}$ ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15

Continuous solutions for balance laws

Abstract: The paper employs the method of characteristics to show that continuous solutions of scalar conservation laws do not incur entropy production and to recover, in a direct and elementary way the (known) properties of solutions to the Hunter-Saxton equation. Keywords: Balance laws, Characteristics, Hunter-Saxton equation Mathematics Subject Classification (2000): 35L65, 35Q58

Criteria for the L p -dissipativity of systems of second order differential equations

Abstract: We give complete algebraic characterizations of the L p -dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathop{\cal A} olimits}^{hk}(x)\partial_{k})$ , where ${\mathop{\cal A} olimits}^{hk}(x)$ are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L p -dissipative if and only if $$ \left({1\over 2}-{1\over p}\right)^{2} \leq {2( u-1)(2 u-1)\over (3-4 u)^{2}}, $$ ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the L p -dissipativity of the operator $\partial_{h} ({\mathop{\cal A} olimits}^{h}(x)\partial_{h})$ , where ${\mathop{\cal A} olimits}^{h}(x)$ are m× m matrices with complex L1loc entries, and we describe the maximum angle of L p -dissipativity for this operator. Keywords: L p -dissipativity, Algebraic conditions, Elasticity system Mathematics Subject Classification (2000): 47D03, 47D06, 47B44, 74B05

A new method to exploit the Entropy Principle and galilean invariance in the macroscopic approach of Extended Thermodynamics

Abstract: In extended thermodynamic the entropy principle and the Galilean invariance dictate respectively constraints for the constitutive equations and the velocity dependence. The entropy principle in particular requires the existence of a privileged field, the main field u′, such that the original system becomes symmetric hyperbolic and is generated by four potentials. It is not easy to solve the restrictions of both principles, if we use as field the non convective main field $\widehat{\mathbf{u}}^{\prime}\ $ and the velocity v. This is due to the fact that $\widehat{\mathbf{u}}^{\prime}$ are not independent. Rather its components satisfy three scalar constraints. The aim of this paper is to solve the full problem using as new strategy to consider $\widehat{\mathbf{u}}^{\prime}\ $ as independent variables and requiring an appropriate differential constraint. This new procedure is very efficient and we are able to solve the problem of 13 moments in the full non linear case (far from equilibrium). It turns out that the knowledge of only the equilibrium state function is sufficient to close the system. Keywords: Extended Thermodynamics, Entropy Principle, Galilean invariance, Rarefied Gas, Hyperbolic systems Mathematics Subject Classification (2000): 74A20, 76P05, 35l60

Nagata's conjecture for a square or nearly-square number of points

Abstract: In this paper we prove that a linear system of plane curves of degree d with n points of multiplicity m always has the expected dimension when n is a perfect square. We also prove some results on the dimension of the linear system when n is close to a perfect square. Keywords: Interpolation, Nagata, Linear systems, Plane curves Mathematics Subject Classification (2000): 14J26, 14N05

Groups with finitely many normalizers of non-abelian subgroups

Abstract: A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation. Keywords: normalizer subgroup, metahamiltonian group Mathematics Subject Classification (2000): 20F24

On a perturbed anisotropic equation with a critical exponent

Abstract: We study a perturbed anisotropic equation without using the knowledge of the limiting problem. This provides a different method from that introduced by Brezis and Nirenberg [4]. Our arguments use some tools recently developed in [5, 6]. Keywords: Anisotropic critical exponent, Critical level, Compactness, Nehari manifold Mathematics Subject Classification (2000): 35J25, 35J60, 35J65

On the principle of virtual powers in continuum mechanics

Abstract: We give a general formulation of the Principle of virtual powers in Continuum Mechanics from a distributional point of view, and study some of its relevant consequences in the field of balance equations. Keywords: Virtual Powers, Contact Interactions, Balance Equations Mathematics Subject Classification (2000): 74A10, 74G70, 74A30, 74A60

Liouville theorems in halfspaces for parabolic hypoelliptic equations

Abstract: We prove some one-side Liouville-type theorems in halfspaces for a class of evolution hypoelliptic equations. The operators we deal with are left trans- lation invariant, and homogeneous of degree two, on homogeneous Lie groups on R N+1 .

Prüfer *-multiplication domains and *-coherence

Abstract: The purpose of this paper is to deepen the study of the Prufer ⋆–mul-tiplication domains, where ⋆ is a semistar operation. For this reason, we introduce the ⋆–domains, as a natural extension of the v-domains. We investigate their close relation with the Prufer ⋆-multiplication domains. In particular, we obtain a characterization of Prufer ⋆-multiplication domains in terms of ⋆–domains satisfying a variety of coherent-like conditions. We extend to the semistar setting the notion of $\texttt{H}$ -domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the $\texttt{H}(\star)$ –domains, the Prufer ⋆-multiplication domains coincide with the ⋆-domains. Keywords: Star and semistar operation, Prufer (⋆-multiplication) domain, $\texttt{H}$ -domain, Localizing system, Coherent domain, Divisorial and invertible ideal Mathematics Subject Classification (2000): 13F05, 13G05, 13E99

Fundamental solution and sharp $L^p$ estimates for Laplace operators in the contact complex of Heisenberg groups

Abstract: In this paper we construct a fundamental solution for the Laplace operator on the contact complex in Heisenberg groups ${\mathbb H}^{n}$ (Rumin’s complex) relying on the notion of currents in ${\mathbb H}^{n}$ given recently by Franchi, Serapioni and Serra Cassano. This operator is of order 2 on k intrinsic forms for k≠ n, but is of order 4 on n intrinsic forms. As an application, we prove sharp L p a priori estimates for horizontal derivatives. Keywords: Heisenberg groups, Differential forms, Currents, Laplace operators, Fundamental solution Mathematics Subject Classification (2000): 43A80, 58A10, 58A25, 35A08

Groups in which the hypercentral factor group is a T-group

Abstract: We consider groups G having a T - group as factor G/Z*(G) and exhibit connections with its Frattini subgroup and its nilpotemt radical.Mutually permutable products of these groups with supersolvable ones are described with consequences concerning the Fitting core of supesolvable groups. Keywords: Hypercenter, T-group, Fitting core Mathematics Subject Classification (2000): 20D25, 20D40, 20D10

Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method

Abstract: The pointwise gradient constrained homogenization process, for Neumann and Dirichlet type problems, is analyzed by means of the periodic unfolding method recently introduced in [21]. Classically, the proof of the homogenization formula in presence of pointwise gradient constraints relies on elaborated measure theoretic arguments. The one proposed here is elementary: it is based on weak convergence arguments in L p spaces, coupled with suitable regularization techniques. Keywords: Homogenization, Gradient constrained problems, Periodic unfolding method Mathematics Subject Classification (2000): 49J45, 35B27, 74Q05

Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials

Abstract: In the present paper we consider for a < x < b, 0 < t < T, the system of partial differential equations $$ \begin{array}{l} \displaystyle{\rho(x) {\partial v \over \partial t} - {\partial \over \partial x} \left(\mu(x,\theta) {\partial v \over \partial x}\right) = f,}\\ \\ \displaystyle{c(x,\theta) {\partial\theta \over \partial t} = \mu(x,\theta) \left({\partial v \over \partial x}\right)^2}, \end{array} $$ completed by boundary conditions on v and by initial conditions on v and θ. The unknowns are the velocity v and the temperature θ, while the coefficients ρ, μ and c are Caratheodory functions which satisfy $$ 0 < c_1 \leq \mu(x,s) \leq c_2, \quad {\partial\mu \over \partial s}(x,s) \leq 0, $$ $$ 0 < c_3 \leq c(x,s) \leq c_4,\quad 0 < c_5 \leq \rho(x) \leq c_6. $$ This one dimensional system is a model for the behaviour of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and temperature dependent rate of plastic work converted into heat. Under the above hypotheses we prove the existence of a solution by proving the convergence of a finite element approximation. Assuming further that μ is Lipschitz continuous in s, we prove the uniqueness of the solution, as well as its continuous dependence with respect to the data. We also prove its regularity when suitable hypotheses are made on the data. These results ensure the existence and uniqueness of one solution of the system in a class where the velocity v, the temperature θ and the stress $\sigma = \mu(x,\theta) \displaystyle{\partial v \over \partial x}$ belong to L∞((0,T) × (a,b)). Keywords: Thermoviscoplastic materials, nonhomogeneous materials, thermal softening, existence, uniqueness, Galerkin’s method Mathematics Subject Classification (2000): 74H20, 74H25, 65M60, 35D05, 35D10, 35R05, 74C10, 74F05, 35Q72, 35M20 This is a “Springer Open Choice” article. Unrestricted non-commercial use, distribution, and reproduction in any medium is permitted, provided the original author and source are credited.

\(\mathfrak{F}\)-critical groups, \(\mathfrak{F}\)-subnormal subgroups, and the generalised Wielandt property for residuals

Abstract: The objective of the present paper is to give a complete answer to some questions concerning $\mathfrak{F}$ -critical groups, associated with a formation $\mathfrak{F}$ , and groups generated by $\mathfrak{F}$ -subnormal subgroups. Keywords: Finite group, Subnormal subgroups, Formations Mathematics Subject Classification (2000): 20D10, 20D20, 20F17

A duality theory for some non-convex functions of matrices

Abstract: We study a special class of non-convex functions which appear in non- linear elasticity, and we prove that they have a well-defined Legendre transform. Several examples are given, and an application to a nonlinear eigenvalue problem.

Reconstructing pluricanonical embeddings of general $k$-gonal curves via odd theta-characteristics

Abstract: Here we investigate r-canonical embeddings of general k-gonal curves of genus g from the point of view of Caporaso–Sernesi’s reconstruction procedure via odd theta-characteristics. Keywords: Theta-characteristic, general k-gonal curve, trigonal curve, pluricanonical embedding, Hilbert scheme Mathematics Subject Classification (2000): 14H50, 14N05

Non complete linear series on curves

Abstract: The paper is concerned with some properties of linear series on smooth plane curves; in fact, we study mainly the case of cubic curves. The main result describes the growth of the dimension of non complete linear series, generalizing to cubics a well known result of Gieseker, about linear series on the projective lines. Keywords: Curves, Linear series Mathematics Subject Classification (2000): 14Q05