Catálogo de publicaciones - libros

We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data.

Measurements of video data on melting dendritic crystal fragments in reduced gravity show that a fragment’s ellipsoidal axial ratio, , rises initially until it melts down to a pole-to-pole length of ≈ 5 mm. At that point we observe a sudden fall in the ratio with time, as the polar regions melt toward each other more rapidly than times the melting speed, , of the equatorial region. This accelerated melting allows the ratio to fall from values around 10–20 (needle-like) towards values approaching unity (spheres) just before total extinction occurs. Analytical and numerical modeling will be presented that suggest that the cause of these sudden changes in kinetics and morphology during melting at small length scales is due to a crystallite’s extreme shape anisotropy. Shape anisotropy leads to steep gradients in the mean curvature of the solid-melt interface near the ellipsoid’s poles. These curvature gradients act through the Gibbs-Thomson effect to induce unusual thermo-capillary heat fluxes that account for the observed enhanced polar melting rates. Numerical evaluation of the thermo-capillary heat fluxes shows that they increase rapidly with the ratio, and with decreasing length scale, as melting progresses toward total extinction.

In this note it is shown that the weak solutions of the Stefan problem for the singular -Laplacian are continuous up to { = 0}. The result is a follow-up to a recent paper of the authors concerning the interior regularity.

We study the existence of positive and sign-changing solutions to the boundary value problem − Δ u = || in a bounded smooth domain Ω in ℝ, with homogeneous Dirichlet boundary condition, when is a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution concentrating at exactly points as →∞. In particular, for a non-simply connected domain such a solution exists for any given ≥ 1. Moreover, for large enough, we prove the existence of two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of Ω.

Surfactants are substances that preferentially accumulate at interfaces between two fluids, altering the local surface tension. An imposed flow can produce a non-uniform distribution of surfactant. In regions of high surfactant concentration the surface tension is low, so the interface offers less resistance to deformation and can become highly curved, allowing very small droplets or bubbles to pinch off. A numerical method to simulate interfacial surfactant mechanics within a volume of fluid method has been developed. To conserve surfactant, the surfactant mass and the interfacial surface area are tracked as the interface evolves, and then the surfactant concentration is reconstructed. The algorithm is coupled to an incompressible flow solver that uses a continuum method to incorporate both the normal and tangential components of the surface tension force into the momentum equation.

We show the American Call on a basket (i.e., a weighted sum of assets) and with dividends increases with the basket volatility in a large class of multivariate continuous diffusion models. In case of a flat yield curve the same result holds for the American Put on a basket. The proof of our result is based on extensions of Hajek’s mean stochastic comparison results to stochastic sums. We provide a simple proof of Hajek’s result and show why the argument is much more involved in case of our extensions. We provide the main ideas of the proofs of our extensions based on heat kernel expansions.

Free boundary problems associated with biological tissue growing under conditions of nutrient limitation are formulated. We consider two constitutive laws to describe the deformation of the tissue, Darcy flow and Stokes flow. Analysis by asymptotic methods, clarifying the model’s stability properties, is then described for the two biologically-plausible limit cases: (1) the small-death-rate limit and (2) the thin-rim (fast-nutrient-consumption or large-tumour) limit. The former leads to some interesting variants on the Hele-Shaw squeeze film problem and the latter makes explicit a buckling instability associated with growth-induced compressive stresses.

A non-smooth temperature-driven phase separation model with conserved energy and a large set of equilibria is shown to develop spontaneously two different time scales as time tends to infinity. The temperature evolution becomes slower and slower, while the microevolution on the unknown phase interface keeps its own independent characteristic speed. In the large time limit, the temperature becomes uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.

The aim of this work is to study obstacle problems associated to monotone operators when the forcing term is a bounded Radon measure. Existence, uniqueness, stability results, and properties of the solutions are investigated.

We apply the Piecewise Constant Level Set Method (PCLSM) to interface problems, especially for elliptic inverse and multiphase motion problems. PCLSM allows using one level set function to represent multiple phases, and the interfaces are represented implicitly by the discontinuity of a piecewise constant level set function. The inverse problem is solved using a variational penalization method with total variation regularization of the coefficient, while the multiphase motion problem is solved by an Additive Operator-Splitting scheme.