The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in...

Let $f$ be an odd function of a class ${\mathrm{C}}^2$ such that $f\left(1\right)=0,f^{\text{'}}\left(0\right)\<0,f^{\text{'}}\left(1\right)\>0$ and $x\mapsto f\left(x\right)/x$ increases on $[0,1]$. We approximate the positive solution of $-\Delta u+f\left(u\right)=0,$ on ${ℝ}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ${]0,L[}^2$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_L-u$ tends to zero exponentially fast, in the uniform norm.

Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f\left(x\right)/x$ increases on [0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on ${}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ]0,L[2 with adequate non-homogeneous Dirichlet conditions. We show that the error uL - u tends to zero exponentially fast, in the uniform norm.

This note deals with the approximation, by a P1 finite element method with numerical integration, of solution curves of a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to H2. So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “a priori adapted” to the singularity. For the H1 or L2-norms, we...

In this paper, we concern ourselves with uniqueness results for an elliptic-parabolic quasilinear partial differential equation describing, for instance, the pressure of a fluid in a three-dimensional porous medium: within the frame of mathematical modeling of the secondary recovery from oil fields, the handling of the component conservation laws leads to a system including such a pressure equation, locally elliptic or parabolic according to the evolution of the gas phase.

We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+u^{7/3}=0$ in $\Omega$, $u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain in ${ℝ}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_K>0$ such that for $0<\mu <{\mu }_K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Let $L$ be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in $L^p$ norm between the maximal function and the square function of solutions to $L$ in Lipschitz domains. Several applications of this result are discussed.

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation ${\square }_{\mathbf{g}}\phi =0$, where $\gg$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor $\mathbf{R}$ of $\gg$ is a major step towards the proof of the bounded $L^2$ curvature conjecture.