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Dynamical approaches to Sharp Sobolev inequalities

Abstract : Sobolev inequalities have been, every since their original proof in 1938, a very active research interest, and have lead to very many applications in various fields of analysis and probability. Notably, they are at the center of the study of partial derivative equations, because they describe a fundamental property of weak derivatives: integrability of a weak derivative translates to strong regularity. Weak derivatives, also known as distributional derivatives, constitute a good setting for the study of PDEs, allowing for instance to make sense of the derivative of a shockwave, a very common physical phenomena. In most applications, knowing the sharp form of an inequality is not particularly better than having estimates of the sharp constants. However, proving sharp inequalities is interesting in itself for at least two reasons. The first one is their link with PDEs, and thus with physical models. Optimal functions are then exactly ground states of the physical problem the model describes. The other reason is the very strong relationship between Sobolev inequalities and the geometry of the underlying space. For example, the isoperimetric inequality in the Euclidean space is equivalent to a special case of the classical Sobolev inequality. The manuscript is divded in 5 chapters. The first two make up the introduction, firt in French and then in English. Concepts and tools used in the other chapters are defined therein, such as some elementary results from the Brunn-Minkowski theory, then from optimal transport, both of those being the starting point of chapter 3. Then, we showcase the Bakry-Émery method, which constitutes an interesting reversal: instead of using Sobolev to study long-term behavior of solutions to certain PDEs, like the heat equation, we use these equations to prove Sobolev inequalities. Chapters 3 to 5 are almost verbatim reproductions of articles written during the thesis. In chapter 3, we use an improved Borell-Brascamp-Lieb inequality, proved in the introduction using optimal transport, to prove new trace inequalities on convex domains of the Euclidean space. The key to this study is the introduction of an infimal convolution. In chapter 4, which was written with Ivan Gentil, Riemannian manifolds are studied. Thanks to the theory of Markov semigroups, weighted Poincaré inequalities as well as weighted Beckner inequalities are proved using solutions to linear parabolic PDEs, with relatively weak hypotheses on the manifolds at hand. Manifolds with negative effective dimension (but with positive Ricci curvature) are an exciting new application of the methods developped in the chapter. Lastly, we construct in chapter 5 an efficient setting to prove sharp Sobolev inequalities in convex domains of the Euclidean space, with applications to manifolds in mind. The equations considered in this chapter are rather different then those in chapter 4, since they are non linear, and with boundary conditions, which renders semigroup thoery completely useless. Despite of that, the results are surprisingly similar to the linear case
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Submitted on : Monday, May 18, 2020 - 5:25:26 PM
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Simon Zugmeyer. Dynamical approaches to Sharp Sobolev inequalities. Functional Analysis [math.FA]. Université de Lyon, 2019. English. ⟨NNT : 2019LYSE1253⟩. ⟨tel-02611806⟩

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