A Ginzburg-Landau type energy with weight and with convex potential near zero

In this paper, we study the asymptotic behaviour of minimizing solutions of a Ginzburg-Landau type functional with a positive weight and with convex potential near $0$ and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis-Merle-Rivi\`ere

A nonlinear problem witha weight and a nonvanishing boundary datum

We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(\Omega),\|u\|_{q}=1} \int_{\Omega}{p(x)}|\nabla{u(x)}|^{2}dx-\lambda\int_{\Omega}| u(x)|^{2}dx$$ where $\Omega$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $p : \bar{\Omega}\longrightarrow \R$ is a given positive weight such that $p\in H^{1}(\Omega)\cap C(\bar{\Omega})$, $0< c_1 \leq p(x) \leq c_2$, $\lambda$ is a real constant and $q=\frac{2n}{n-2}$ and $g$ a given positive boundary data. The goal of this present paper is to show that minimizers do exist. We distinguish two cases, the first is solved by a convex argument while the second is not so straightforward and will be treated using the behavior of the weight near its minimum and the fact that the boundary datum is not zero

Solutions positives de l'équation −Δu=up+μuq- \Delta u = u^p + \mu u^q dans un domaine à trou

International audienc

Regularity of minimizing maps with values in S2\mathbb{S}^2 and some numerical simulations

International audienc

A NONLINEAR PROBLEM WITH A WEIGHT AND A NONVANISHING BOUNDARY DATUM

International audienc