The paper considers the static Maxwell system for a Lipschitz domain with perfectly conducting boundary. Electric and magnetic permeability ε and μ are allowed to be monotone and Lipschitz continuous functions of the electromagnetic field. The existence theory is developed in the framework of the theory of monotone operators.

On considère le problème de Dirichlet :$$(d{d}^{c}u{)}^n=0\mathrm{dans}B$$et$$u{|}_{\partial B}=\varphi$$où $B$ désigne la boule unité de ${ℂ}^n.$Nous donnons une démonstration simple du fait que si $\varphi \in C^{1,1}\left(\partial B\right)$, alors $u\in C^{1,1}\left(B\right)$; de plus la croissance du coefficient de Lipschitz de la différentielle de $u$ est contrôlée par l’inverse de la distance au bord.

Ce papier porte sur l’étude mathématique d’une équation du type de Grad-Mercier qui décrit, dans certaines circonstances, l’équilibre d’un plasma confiné [H. Grad, P.N. Hu et D.C. Stevens, Proc. Nat. Acad. Sci. USA, 72,n${}^{\circ }$10 (1975), 3789–3793, C. Mercier, Publication of Euratom, CEA, Luxembourg (1974), C. Mercier, Communications personnelles à R. Temam et aux auteurs]. Il s’agit de trouver une fonction “régulière” $u$ solution du système$$\left\{\begin{array}{cc}-\Delta u+\lambda g\left[\delta \right(u\left)\right]=0\hfill & \text{dans}\Omega ,\hfill \\ u=\text{constante}\text{(inconnue)}\>0\hfill & \text{sur}\partial \Omega ,\hfill \\ {\int }_{\partial \Omega }\frac{\partial u}{\partial n}=I,\hfill \end{array}\right.$$où $\Omega$ est un ouvert borné régulier de ${\mathbf{R}}^n$, et$$\delta \left(u\right)\left(x\right)=\mathrm{mes}\{y\in \Omega \mid u(x)\<u(y)\<0\}.$$L’opérateur non linéaire...

We study uniformly elliptic fully nonlinear equations $F(D^{2}u,Du,u,x)=0$, and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

Le but de l’exposé est de présenter les résultats obtenus par S. Bianchini et A. Bressan sur le problème de Cauchy pour des perturbations visqueuses ${\partial }_t{u}^{\epsilon }+{\partial }_{x}f\left(u^{\epsilon }\right)=\epsilon {\partial }_{xx}{u}^{\epsilon }$ de systèmes strictement hyperboliques ${\partial }_{t}u+{\partial }_{x}f\left(u\right)=0$ en une dimension d’espace. Ils ont en particulier montré l’existence globale ($t\ge 0$), l’unicité et la stabilité des solutions et justifié la convergence quand $\epsilon$ tend vers zéro pour des données initiales à petite variation totale. Leur analyse montre aussi que les solutions du système hyperbolique ainsi obtenues...

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of...

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

In questa nota viene introdotto un nuovo metodo per ottenere espressioni esplicite dell'energia della soluzione dell'equazione iperbolica $${\left(\frac{\partial }{\partial t}\right)}^{m}u+\sum _{|\nu |+j\le m;j\le m-1}{a}_{\nu ,j}(t){\left(\frac{\partial }{\partial x}\right)}^{\nu }{\left(\frac{\partial }{\partial t}\right)}^{j}u=0.$$ Stimando opportunamente queste espressioni si ottengono nuovi risultati di buona positura negli spazi di Gevrey per l'equazione $(\cdot )$ quando questa è debolmente iperbolica.

In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.