In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in $S_{1,0}^2$ such that $\left|a\right|\le b$, then for all $ϵ\>0$ we have the estimate $\parallel a^{w}u{\parallel }_s^2\le C_{ϵ}(\parallel b^{w}u{\parallel }_s^2+\parallel u{\parallel }_{s+ϵ}^2)$ for all $u$ in the Schwartz space, where $\parallel {\parallel }_t$ is the usual $H_t$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

We consider a fully practical finite element approximation of the following degenerate system$$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi$. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)\>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

We consider a fully practical finite element approximation of the following degenerate system $$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$ subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step...

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface...

In this work the problem of identificating flaws or voids in elastic solids is addressed both from a theoretical and an experimental point of view. Following a so called inverse procedure, which is based on appropriately devised experiments and a particular bounding of the strain energy, a gap functional for flaw identification is proposed.

We study the existence and the uniqueness of a solution $\varphi$ to the linear Fokker-Planck equation $-\Delta \varphi +div\left(\varphi \mathbf{F}\right)=f$ in a bounded domain of ${ℝ}^d$ when $\mathbf{F}$ is a “confinement” vector field. This field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.

The paper is concerned with the graph formulation of forced anisotropic mean curvature flow in the context of the heteroepitaxial growth of quantum dots. The problem is generalized by including anisotropy by means of Finsler metrics. A semi-discrete numerical scheme based on the method of lines is presented. Computational results with various anisotropy settings are shown and discussed.

Certain hyperbolic equations with continuous distributed deviating arguments are studied, and sufficient conditions are obtained for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities by using some integral means of solutions.