Solution of the Monge-Ampère equation on Wiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge-Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400-428], w.r.t the Wiener measure is also the strong solution of the Monge-Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.     

The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian

For a class of elliptic Hessian operators raised by Caffarelli-Nirenberg-Spruck, the corresponding parabolic Monge-Ampère equation was studied, the existence and uniqueness of the admissible solution to the first initial-boundary value problem for the equation were established, which extended a result of Ivochkina-Ladyzhenskaya.     

Settling motion of interacting solid particles in the vicinity of a plane solid boundary

The sedimentation of $N?1$ small arbitrarily-shaped solid bodies near a solid plane is addressed by discarding inertial effects and using 6N boundary-integral equations. Numerical results for 2 or 3 identical spheres reveal that combined wall–particle and particle–particle interactions deeply depend on the cluster's geometry and distance to the wall and may even cancel for a sphere which then moves as it were isolated. To cite this article: A. Sellier, C. R. Mecanique 333 (2005).     

Un critère de Darboux d'existence d'intégrale première pour les 1-formes différentielles analytiques

Resumé On considère une 1-forme analytique définie sur un ouvert borné de ⁿ et qui s'étend au voisinage de l'adhérence de cet ouvert. On montre que si elle possède une infinité de feuilles contenues dans des hypersurfaces algébriques différentes et de degrés bornés alors elle admet une intégrale première méromorphe. Si de plus l'origine est dans l'adhérence de chacune de ces feuilles alors la 1-forme admet une intégrale première méromorphe de graphe algébrique. Des exemples illustrent l'optimalité de ces résultats.

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Consider a germ of analytic differential one form and suppose there exists infinitely separatrix in algebraic surfaces of bounded degree. The conclusion is that there exists an algebraic first integral. We give a list of examples wich explain the necessity of the hypotheses.

Financements: CNRS et réseau européen TMR Sing. Eq. Diff. et Feuilletages     

Hypersurfaces with Prescribed Affine Gauss–Kronecker Curvature

We extend a procedure for solving particular fourth order PDEs by splitting them into two linked second order Monge–Ampère equations. We use this for the global study of Blaschke hypersurfaces with prescribed Gauss–Kronecker curvature.     

Atomic‐Scale Visualization of the Stepwise Metal‐Mediated Dehalogenative Cycloaddition Reaction Pathways: Competition between Radicals and Organometallic Intermediates

Dehalogenative cycloaddition reaction is a powerful strategy to generate new ring scaffolds with π‐conjugated features on a surface, and thus holds great promise toward atomically precise electronic devices or nanomaterials. The ortho‐dihalo substitution provides a good strategy to realize cycloaddition. However, the limited understanding of intermediate states involved hinders mechanistic exploration for further precise design and optimization of reaction products. Now, the evolutions of competing surface‐stabilized radicals and organometallic intermediates in real space were visualized toward the formation of dominant conjugated four‐membered ring connections. From the interplay of scanning tunneling microscopy and density functional theory calculations, the stepwise metal‐mediated dehalogenative cycloaddition pathway is elucidated both experimentally and theoretically. The results provide fundamental insights into the intermediate states involved in on‐surface synthesis.     

Enantioselective Synthesis of Acyclic α‐Quaternary Carboxylic Acid Derivatives through Iridium‐Catalyzed Allylic Alkylation

The first highly enantioselective iridium‐catalyzed allylic alkylation that provides access to products bearing an allylic all‐carbon quaternary stereogenic center has been developed. The reaction utilizes a masked acyl cyanide (MAC) reagent, which enables the one‐pot preparation of α‐quaternary carboxylic acids, esters, and amides with a high degree of enantioselectivity. The utility of these products is further explored through a series of diverse product transformations.     

Singular Solutions to Monge-Ampère Equation

We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side.