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The Young inequality and the Δ₂-condition

Philippe Laurençot — 2002

Colloquium Mathematicae

If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality x y ε φ ( x ) + C ε φ * ( y ) is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

Hysteresis filtering in the space of bounded measurable functions

Pavel KrejčíPhilippe Laurençot — 2002

Bollettino dell'Unione Matematica Italiana

We define a mapping which with each function u L 0 , T and an admissible value of r > 0 associates the function ξ with a prescribed initial condition ξ 0 which minimizes the total variation in the r -neighborhood of u in each subinterval 0 , t of 0 , T . We show that this mapping is non-expansive with respect to u , r and ξ 0 , and coincides with the so-called play operator if u is a regulated function.

A stochastic min-driven coalescence process and its hydrodynamical limit

Anne-Laure BasdevantPhilippe LaurençotJames R. NorrisClément Rau — 2011

Annales de l'I.H.P. Probabilités et statistiques

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

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