Catálogo de publicaciones - libros

In our previous works we have proposed a mathematical model for dynamics of shape memory alloy materials. In the model the relationship between the strain and the stress is given as the generalized stop operator described by the ordinary differential equation including the subdifferential of the indicator function for the closed interval depending on the temperature. Here, we adopt the Duhem type of hysteresis operators as the mathematical description of the relationship in order to deal with the more realistic mathematical model. The aims of this paper are to show our new model and to establish the well-posedness of the model.

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a()+() ∋ , posed in an open bounded subset Ω of ℝ, with nonlinear boundary conditions of the form a()·+() ∋ . The nonlinear elliptic operator div a() is modeled on the -Laplacian operator Δ() = div (|Du|), with > 1, and are maximal monotone graphs in ℝ such that 0 ∈ (0) and 0 ∈ (0), and the data ∈ (Ω) and ∈ (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem.

In this work we consider an incompressible, non-homogeneous, dilatant and viscous fluid for which the stress tensor satisfies a general non-Newtonian law. The new contribution of this work is the consideration of an anisotropic dissipative forces field which depends nonlinearly on the own velocity. We prove that, if the flow of such a fluid is generated by the initial data, then in a finite time the fluid becomes immobile. We, also, prove that, if the flow is stirred by a forces term which vanishes at some instant of time, then the fluid is still for all time grater than that and provided the intensity of the force is suitably small.

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional ()-Laplacian equation. We prove the existence of a bounded weak solution and study its localization (vanishing) properties.

We consider a system of nonlinear parabolic PDEs which includes a constraint on the time-derivative depending on the unknowns. This system is a mathematical model for irreversible phase transitions. In our phase transition model, the constraint := () is a function of the temperature and the order parameter (state variable) and it is imposed on the velocity of the order parameter, for instance, in such a way that () ≤ ≤ () + (a positive constant). We give an existence result of the problem.

We consider the problem of finding the equilibrium position of membranes constrained not to pass through each other, under prescribed volumic forces and boundary tensions. This model corresponds to solve variationally a -system for linear second order elliptic equations with sequential constraints. We obtain interior and boundary Lewy-Stampacchia type inequalities for the respective solution and we establish the conditions for stability in measure of the interior contact zones of the membranes.

In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, chemical and biological processes of contaminant transport and degradation in the subsurface. Our latest results for the existence, uniqueness and regularity of solutions to the model equations are summarized; cf. [, ]. The basic idea of the proof of regularity is sketched briefly. Moreover, our numerical discretization scheme that has proved its capability of approximating reliably and efficiently solutions of the mathematical model is described shortly, and an error estimate is given; cf. [, ]. Finally, to illustrate our approach of modelling and simulating bioreactive transport in the subsurface, the movement and expansion of a -xylene plume is studied numerically under realistic field-scale assumptions.

We consider the Nitzberg-Mumford variational formulation of the segmentation with depth problem. This is an image segmentation model that allows regions to overlap in order to take into account occlusions between different objects. The model gives rise to a variational problem with free boundaries. We discuss some qualitative properties of the Nitzberg-Mumford functional within the framework of the relaxation methods of the Calculus of Variations. We try to characterize minimizing segmentations of images made up of smooth overlapping regions, when the weight of the fidelity term in the functional becomes large. This should give some theoretical information about the capability of the model to reconstruct both occluded boundaries and depth order.

We review some results about the first eigenvalue of the infinity Laplacian operator and its first eigenfunctions in a general norm context. Those results are obtained in collaboration with several authors: V. Ferone, P. Juutinen and B. Kawohl (see , , and ). In Section 5 we make some remarks on the simplicity of the first eigenvalue of Δ: this will be the object of a joint work with A. Wagner (see ).

Amerasian options pricing problems are formulated, using Black-Scholes and Merton methodology, as unilateral obstacle problems for degenerate parabolic convection-diffusion-reaction operators. We mainly focus on the numerical solution of these problems and we compare two algorithms based on the augmented Lagrange formulation. Moreover, we use higher-order Lagrange-Galerkin methods for the time-space discretization. Finally, numerical results show the performance of the proposed methods.