A review of some methods in sheaf theory is presented to make precise a general concept of regularity in algebras or spaces of generalized functions. This leads to the local analysis of the sections of sheaves or presheaves under consideration and then to microlocal analysis and microlocal asymptotic analysis.

Nella prima parte di questa Nota si dimostrano dei risultati di simmetria unidimensionale e radiale per le soluzioni di $\mathrm{\Delta }u+f\left(u\right)=0$ in ${\mathbb{R}}^N$. Questi risultati sono legati a due congetture (De Giorgi, 1978 e Gibbons, 1994) riguardanti la classificazione delle soluzioni dell’equazione $\mathrm{\Delta }u+u\left(1-u^2\right)=0$ in ${\mathbb{R}}^N$. Si dimostra, in particolare, la seguente generalizzazione della congettura di Gibbons: se $N>1$ e se l’insieme degli zeri di $u$ è limitato nella direzione $\nu$, allora $u\left(x\right)=u_0\left(\nu \cdot x\right)$, ovvero, $u$ è unidimensionale. Nella seconda parte si considerano...

When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations...

When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations...

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small...

Étant donné $L$ un opérateur différentiel d’ordre $m$ sur un ouvert $\Omega$ de ${\mathbf{R}}^N$, $K$ un compact de $\Omega$, $\gamma \>1$ et ${\gamma }^{\text{'}}=\gamma /(\gamma -1)$, nous montrons que toute solution de “$Lu+u^{\gamma }=0$ sur $\Omega \setminus K,u\ge 0$” est solution de “$Lu+u^{\gamma }=0$ sur $\Omega$” dès que la $W^{m,{\gamma }^{\text{'}}}$-capacité de $K$ est nulle. Cette condition s’avère nécessaire quand $L$ est un opérateur elliptique d’ordre 2. Dans ce cas, nous montrons aussi que $``Lu+u|u{|}^{\gamma -1}=\mu ,u{|}_{\partial \Omega }=0^{\text{'}\text{'}}$ où $\mu$ est une mesure de Radon bornée sur $\Omega$, a une solution si et seulement si $\mu$ ne charge pas les ensembles de $W^{2,{\gamma }^{\text{'}}}$-capacité nulle.

We study the regularity of the solution of the regularized electric Maxwell problem in a polygonal domain with data in $L^p{\left(\Omega \right)}^2$. Using a duality method, we prove a decomposition of the solution into a regular part in the non-Hilbertian Sobolev space $W^{2,p}{\left(\Omega \right)}^2$ and an explicit singular one.