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Approximation of viscosity solution by morphological filters

Denis Pasquignon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider in 2 all curvature equation u t = | D u | G ( curv ( u ) ) where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respect to any contrast change u → g(u), with g nondecreasing. Consider the contrast invariant operator T t : u o u ( t ) . A Matheron theorem asserts that all contrast invariant operator T can be put in a form ( T u ) ( 𝐱 ) = inf B sup 𝐲 B u ( 𝐱 + 𝐲 ) . We show the asymptotic equivalence of both formulations. More precisely, we show that all curvature equations can be obtained...

Around the bounded L 2 curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel (2008)

Journées Équations aux dérivées partielles

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation g φ = 0 , where is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L 2 bounds on the curvature tensor R of is a major step towards the proof of the bounded L 2 curvature conjecture.

Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature

Shigetoshi Yazaki (2007)

Kybernetika

Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex:...

Asymptotic behaviour and the moduli space of doubly-periodic instantons

Olivier Biquard, Marcos Jardim (2001)

Journal of the European Mathematical Society

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line , with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T × 1 . The converse statement is also true, namely a holomorphic bundle on T × 1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....

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