Laboratoire de Mathematiques et Physique Theorique Faculte des Sciences et Techniques Universite de Tours
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Abstract:
In a celebrated paper motivated by applications to image analysis, L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel showed that any monotone semigroup dened on the space of bounded uniformly continuous functions, which satises suitable regularity and locality assumptions is in fact a semigroup associated to a fully nonlinear, possibly degenerate, second-order parabolic partial dierential equation. In this paper, we extend this result by weakening the assumptions required on the semigroup to obtain such a result and also by treating the case where the semigroup is dened on a general space of continuous functions like, for example, a space of continuous functions with a prescribed growth at innity. These extensions rely on a completely dierent proof using in a more central way the monotonicity of the semigroup and viscosity solutions methods. Then we study the consequences on the partial dierential equation of various additional assumptions on the semigroup. Finally we brie y present the adaptation of our proof to the case of two-parameters families. Dans un clbre article motiv par les applications au traitement d'image, L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel ont montr qu'un semi-groupe monotone dni sur
