If $L(t,x,{\partial }_t,{\partial }_x)$ is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets ${\Omega }_0\subset \{t=0\}$. The frozen constant coefficient operators $L(\underline{t},\underline{x},{\partial }_t,{\partial }_x)$ determine local convex propagation cones, ${\Gamma }^{+}(\underline{t},\underline{x})$. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points $\Omega$ which cannot be reached by influence curves beginning in the exterior of ${\Omega }_0$ is a domain of...

We consider the singularly perturbed set of periodic functional-differential matrix Riccati equations, associated with a periodic linear-quadratic optimal control problem for a singularly perturbed delay system. The delay is small of order of a small positive multiplier for a part of the derivatives in the system. A zero-order asymptotic solution to this set of Riccati equations is constructed and justified.

La résolution d’un système d’EDP non linéaires, de type mixte et sous contraintes, est étudiée dans des ouverts non bornés. Le cas considéré est celui d’un modèle d’écoulement transsonique avec condition d’entropie. Le problème est ramené à l’annulation d’une fonctionnelle positive pénalisée, dans un cadre hilbertien. Des solutions généralisées à $\epsilon$ près sont obtenues par encadrement de la borne inférieure de la fonctionnelle. Si les contraintes sont omises et sous certaines hypothèses, un algorithme...

Dans cette note, nous prouvons l’existence de solutions indéfiniment différentiables d’un système de deux équations aux différences et appliquons la technique utilisée à l’étude des systèmes d’équations linéaires aux dérivées partielles.Dans chaque cas, on montre que les solutions sont les premières composantes des solutions d’un système matriciel que nous étudions.

In this paper are examined some classes of linear and non-linear analytical systems of partial differential equations. Compatibility conditions are found and if they are satisfied, the solutions are given as functional series in a neighborhood of a given point (x = 0).

We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with ${\int }_{\Omega }f=0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_w^{k,q}\left(\Omega \right)$, where the weight function w is in the class of Muckenhoupt weights $A_q$.

This is the expanded text of a lecture about viscosity solutions of degenerate elliptic equations delivered at the XVI Congresso UMI. The aim of the paper is to review some fundamental results of the theory as developed in the last twenty years and to point out some of its recent developments and applications.

We study the solvability of equations associated with a complex vector field $L$ in ${ℝ}^2$ with $C^{\infty }$ or $C^{\omega }$ coefficients. We assume that $L$ is elliptic everywhere except on a simple and closed curve $\Sigma$. We assume that, on $\Sigma$, $L$ is of infinite type and that $L\wedge \overline{L}$ vanishes to a constant order. The equations considered are of the form $Lu=pu+f$, with $f$ satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of $L\wedge \overline{L}$ is $\>1$, the equation $Lu=f$ is solvable in the $C^{\infty }$ category but not in the $C^{\omega }$ category....

This paper presents a numerical approach to solve the Hamilton-Jacobi-Bellman (HJB) problem which appears in feedback solution of the optimal control problems. In this method, first, by using Chebyshev pseudospectral spatial discretization, the HJB problem is converted to a system of ordinary differential equations with terminal conditions. Second, the time-marching Runge-Kutta method is used to solve the corresponding system of differential equations. Then, an approximate solution for the HJB problem...

Using Clifford analysis in a multidimensional space some elliptic, hyperbolic and parabolic systems of partial differential equations are constructed, which are related to the well-known classical equations. To obtain parabolic systems Clifford algebra is modified and some corresponding differential operator is constructed. For systems obtained the boundary and initial value problems are solved.

This text is the act of a talk given november 18 2008 at the seminar PDE of Ecole Polytechnique. The text is not completely faithfull to the oral exposition for I have taken this opportunity to present the proofs of some results that are not easy to find in the literature. On the other hand, I have been less precise on the material for which I found good references. Most of the novelties presented here come from a joined work with Luigi Ambrosio.