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Theses

Analyse et rectifiabilité dans les espaces métriques singuliers

Abstract : In this thesis, we essentially prove the Cheeger-differentiability of some Hajlasz-Sobolev functions between PI metric spaces and RNP Banach spaces. Then, we prove a refinement. More precisely, we establish a kind of Rademacher-Stepanov Theorem in the same setting as above but under the simple condition that the upper lipschitz constant is in a Lp space. Then, all these differentiation Theorems are naturally used to give a precise and complete description of the Hajlasz-Sobolev spaces on PI metric spaces in term of an energy integral. This leads to some criteria to detect if a measurable function is constant or not. At the end, we discuss some topological consequences of some weak Poincaré inequalities, we mean that depend of the center and of the radius of the balls involved in these inequalities. In this context, we are able to give some new criteria but the price to pay is to suppose strong topological assumptions on the metric space.
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Submitted on : Monday, October 10, 2011 - 2:52:23 PM
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Vincent Munnier. Analyse et rectifiabilité dans les espaces métriques singuliers. Mathématiques générales [math.GM]. Université de Grenoble, 2011. Français. ⟨NNT : 2011GRENM033⟩. ⟨tel-00630615⟩

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