We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C^1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

A famous theorem of Harish-Chandra asserts that all invariant eigendistributions on a semisimple Lie group are locally integrable functions. We show that this result and its extension to symmetric pairs are consequences of an algebraic property of the holonomic $\mathcal{D}$-module defined by Hotta and Kashiwara.

Let $V$ be a compact Kähler manifold. Let $\mathcal G'$ be a commutative subgroup of $\Aut(V)$, and let $U$ be the set of elements of zero entropy of $\mathcal{G}'$. Then $U$ is a group and $\mathcal{G}'$ is isomorphic to the direct product $\mathcal{G}'\simeq U\times \mathcal{G}$, where $\mathcal{G}$ is a subgroup of $\mathcal{G}'$ such that all elements of $\mathcal{G}\setminus\id$ are of positive entropy. Moreover, $\mathcal{G}$ is a free commutative group with $\mbox{rank}(\mathcal{G})\leq \dim V-1$. The estimate is sharp. When $\mbox{rank}(\mathcal{G})=\dim V-1$, $U$ is finite. $\mbox{Rank}(\mathcal{G})$ satisfies other inequalities involving the dimensions of Dolbeault cohomology groups of $V$.

Soit $V$ une variété kählérienne compacte et soit $\mathcal{G}'$ un sous-groupe commutatif de $\Aut(V)$. Notons $U$ l'ensemble des éléments d'entropie nulle de $\mathcal{G}'$. Alors $U$ est un groupe. Il existe un sous-groupe $\mathcal{G}$ de $\mathcal{G}'$ dont les éléments différents de l'identité sont d'entropie positive et tel que $\mathcal{G}'$ soit isomorphe au produit $U\times \mathcal{G}$. Le groupe $\mathcal{G}$ est un groupe commutatif libre de rang $r\leq \dim V-1$. L'estimation est optimale. Lorsque $r=\dim V-1$, $U$ est fini. Le rang de $\mathcal{G}$ satisfait d'autres inégalités faisant intervenir la dimension des groupes de cohomologie de Dolbeault de $V$.

We prove that the singular set $S(\mathscr{L})$ of convergence in the Colding-Minicozzi limit lamination theorem is a $C^{1,1}$-curve that is orthogonal to the limit minimal foliation $\mathscr{L}$ in some neighborhood of $S(\mathscr{L})$.

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra. Let $\mathbf{U}^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\mathfrak{g}$. We construct a basis of $\mathbf{U}^+$ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper-triangular matrix (with respect to an explicitly defined ordering) with 1's on the diagonal and with above-diagonal entries in $q_s^{-1} \mathbf{Z}[q_s^{-1}]$. Using this construction, we study the global crystal basis $\mathscr{B}(\widetilde{\mathbf{U}})$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl-like decomposition of the crystal $\mathscr{B}(\widetilde{\mathbf{U}})$ (Th. 4.18), as well as an explicit description of two-sided cells of $\mathscr{B}(\widetilde{\mathbf{U}})$ and the limit algebra of $\widetilde{\mathbf{U}}$ at $q=0$ (Th. 6.44).

We show the necessity of the nontrapping condition for the plain smoothing effect ($H^{1/2}$) for the Schrödinger equation with Dirichlet boundary conditions in exterior problems. We also give a class of trapped obstacles (Ikawa's example) for which we can prove a weak ($H^{1/2 - \varepsilon}$) smoothing effect.

On démontre que l'hypothèse de non capture est nécessaire pour l'effet régularisant ($H^{1/2}$) pour l'équation de Schrödinger avec conditions aux limites de Dirichlet à l'extérieur d'un domaine de $\mathbb{R}^d$. On donne aussi une classe d'obstacles captifs (l'exemple d'Ikawa) pour lesquels on démontre un effet régularisant affaibli ($H^{1/2 - \varepsilon}$).