The aim of this paper is to analyze the well posedness of the one-phase quasi-stationary Stefan problem with the Gibbs-Thomson correction in a two-dimensional domain which is a perturbation of the half plane. We show the existence of a unique regular solution for an arbitrary time interval, under suitable smallness assumptions on initial data. The existence is shown in the Besov-Slobodetskiĭ class with sharp regularity in the L₂-framework.

Vengono brevemente studiati i problemi di Stefan su «capacità concentrate»,seguendo l'approccio recentemente introdotto di G. Savaré e A. Visintin.

We study the generalized Stokes resolvent equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega ₀={ℝ}^{n-1}\times (-1,1)$. Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. We discuss the results on unique solvability of the generalized Stokes resolvent equations as well as the existence of a bounded $H_{\infty }$-calculus for the associated Stokes operator and some...

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...