For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.[-]

We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get rapid stabilization for Korteweg-de Vries equations and how to stabilize in finite time $1-D$ parabolic linear equations by means of periodic time-varying feedback laws.[-]

actions commensurantes # algèbre de Steenrod # algèbres de Lie semi-simple # bases canoniques # biparti # caractère # carte # cartes de mots # catégorification # classification # cohomologie étale # cohomologie galoisienne # cohomologie motivique # commutateurs # complexe des courbes # conjecture de Baum-Connes # conjecture de Bloch-Kato # conjecture de Hodge # conjecture d'Ore # conjecture de Thompson # constantes de Siegel-Veech # corps d'Okounkov # courbure # cycles algébriques # déterminant du laplacien diagramme de Young # différentielles holomorphes # dimension d'Iitaka # distance de Wasserstein # dynamique symbolique # échanges d'intervalles # ÉDP d'évolution # ÉDP stochastiques # endoscopie tordue # équations F-KPP # espace de modules de différentielles quadratiques # espaces métriques mesurés # exposants de Lyapunov # extrêmes # flot de la chaleur # flot géodésique de Teichmüller # flots de gradient # fonction de Hilbert # fonctorialité # formes automorphes de carré intégrable # graphe expanseur # groupe hyperbolique # groupes approximativement finis # groupes classiques # groupes élémentairement moyennables # groupes kleiniens # groupes moyennables # groupes pleins-topologiques # groupes quantiques # homéomorphismes minimaux # hyperbolicité au sens de Kobayashi # inégalités de Morse holomorphes # KK-théorie # K-théorie de Milnor # laminations terminales # mouvement brownien branchant # odomètres # partition # polynôme de Kerov propriété (T) # renormalisation # sous-décalages topologiques # surfaces plates # symétriseur de Young # théorie homotopique des schémas # trajectoires rugueuses # unicellulaire # variations de structure de Hodge [-]