Cet exposé présente les résultats de l’article [3] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.

In this paper we consider a model of a one-dimensional body where strain depends on the history of stress. We show local existence for large data and global existence for small data of classical solutions and convergence of the displacement, strain and stress to zero for time going to infinity.

A mathematical model of heat and mass transport in non-isothermal partially saturated oil-wax solution was formulated by A. Fasano and M. Primicerio [1]. This paper is devoted to the study of a one-dimensional problem in the framework of that model. The existence of classical solutions in a small time interval is proved, based on the application of a fixed-point theorem to the constructed operator. The technique employed is close to the one of [3] and [4].

In this paper we consider two-dimensional quasilinear equations of the form $\text{div}\left(a\left(\left|\nabla u\right|\right)\nabla u\right)+f\left(u\right)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u$(notice that $\text{arg}\left(\nabla u\right)$ is a well-defined real function since $\left|\nabla u\right|>0$ on ${\mathbb{R}}^2$) we prove that $u$ is one-dimensional, i.e., $u=u\left(\nu \cdot x\right)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides a proof of...

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation$$\Delta u+\rho \left(\frac{e^u}{{\int }_T{e}^u}-\frac{1}{\left|T\right|}\right)=0,{\int }_{T}u=0.$$It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8\pi$. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting ${\lambda }_1\left(T\right)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le min\{8\pi ,{\lambda }_1\left(T\right)|T\left|\right\}$. Our results hold more generally for solutions that are Steiner symmetric, up to a translation....

The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra ${}_C^{\text{'}}(ℝⁿ;M_{m\times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G{\in }_C^{\text{'}}(ℝⁿ;M_{m\times m})$ is the generating distribution of an i.d.c.s. if and only if the operator ${\partial }_t{\otimes }_{m\times m}-G\ast$ on ${ℝ}^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...