Advanced Search

Let $W$ be a non-negative function of class ${\mathrm{C}}^3$ from ${ℝ}^2$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let $S^1(𝐚,𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy $$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$ is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_1$ over $S^1(𝐚,𝐛)$. Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at $z_{+}$ and...

Let be a non-negative function of class C from ${}^2$ to $$, which vanishes exactly at two points and . Let (, ) be the set of functions of a real variable which tend to at -∞ and to at +∞ and whose one dimensional energy $$E_1\left(v\right)={\int }_{\left[W\left(v\right)+{\left|v^{\text{'}}\right|}^2/2\right]}x$$ is finite. Assume that there exist two isolated minimizers and of the energy over (, ). Under a mild coercivity condition on the potential and a generic spectral condition on the linearization...

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.

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is $1/2$ in general and $1$ when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl’s rheological model, our estimates in maximum norm do not depend on spatial dimension.

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

In this article, we show the convergence of a class of numerical schemes for certain maximal monotone evolution systems; a by-product of this results is the existence of solutions in cases which had not been previously treated. The order of these schemes is in general and when the only non Lipschitz continuous term is the subdifferential of the indicatrix of a closed convex set. In the case of Prandtl's rheological model, our estimates in maximum norm do not depend on spatial dimension.