Oscillations fortes sur un champ linéairement dégénéré
Partial differential analogs of ordinary differential equations and systems.
Problema di trasporto e equazione di Cauchy per campi vettoriali a variazione limitata
In questa conferenza descrivo alcuni recenti sviluppi relativi al problema dell'unicità per l'equazione differenziale ordinaria e per l'equazione di continuità per campi vettoriali debolmente differenziabili. Descrivo infine un'applicazione di questi risultati a un sistema di leggi di conservazione.
Problème d'obstacle pour une équation quasi-linéaire du premier ordre
Problèmes de Cauchy singuliers non linéaires
Problèmes de convergence dans certaines EDP non linéaires et applications géométriques
Propagation of weak singularities on characteristic surfaces of non-constant multiplicity
Quasiconvex functions and null Lagrangians in the stability problems of classes of mappings.
Quasilinear Resolutions of Non-Linear Equations.
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity
We formulate an Hamilton-Jacobi partial differential equationon a dimensional manifold , with assumptions of convexity of and regularity of (locally in a neighborhood of in ); we define the “min solution” , a generalized solution; to this end, we view as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about ; in particular, we prove in the first part that the closure of the set where is not regular may be covered by a countable number...
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity
We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where...
Regularity of solutions of the Hamilton-Jacobi equation
Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry
Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...
Remarques sur des résultats d'existence pour les équations de Hamilton-Jacobi du premier ordre
Résurgence d'un thème de Huygens-Fresnel
Sharp Domains of Determinacy and Hamilton-Jacobi Equations
If 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 . The frozen constant coefficient operators determine local convex propagation cones, . Influence curves are curves whose tangent always lies in these cones. We prove that the set of points which cannot be reached by influence curves beginning in the exterior of is a domain of...
Solutions à près de systèmes d’équations aux dérivées partielles non linéaires de type mixte posés sur des ouverts non bornés
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 près sont obtenues par encadrement de la borne inférieure de la fonctionnelle. Si les contraintes sont omises et sous certaines hypothèses, un algorithme...
Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique
Solutions of Analytical Systems of Partial Differential Equations
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).
Solutions of DEs and PDEs as potential maps using first order Lagrangians.