报告题目：Entropy methods and stability in functional inequalities

报告摘要：Some functional inequalities of Sobolev type play a crucial role in the study of nonlinear diffusion equations like the fast diffusion equation. As a typical example, optimal time-dependent estimates for entropies are equivalent to optimal constants in some Gagliardo-Nirenberg inequalities. Reciprocally the fast diffusion equations can be used as a tool to investigate the set of optimal functions and the stability properties of these functions in strong norms using various extensions of the carré du champ method. Although limited to simple inequalities with good homogeneity and invariance properties, entropy methods do not rely on symmetrization and offer an alternative approach that can be used, for instance, to establish symmetry results and rigidity properties of the optimal functions in Caffarelli-Kohn-Nirenberg inequalities. The key of these method are the monotonicity properties of entropies, deficit and related functionals, which allow us to reduce a global problem to a local minimization issue that can be handled with spectral methods and (sometimes delicate) Taylor expansions. The goal of the lectures will be to give an overview of the strategies based on various examples and reduce the proofs to elementary questions that are easily accessible.

报告人简介：Jean Dolbeault，法国国家科研中心（CNRS）主任研究员（法国最高等级Directeur de Recherches, classe exceptionnelle），任职巴黎多菲纳大学CEREMADE，是法国泛函不等式与非线性扩散方程领域领军人物之一。曾任 Colloquium Mathematicum 编委（2008–2013）及开放期刊 Ars Inveniendi Analytica 编委，导师 P.-L. Lions（1994 年菲尔兹奖得主）。2000年获Habilitation。 在Invent. Math.、J. Eur. Math. Soc.、J. Math. Pures Appl.等国际顶尖期刊发表论文100多篇。与del Pino求得 Gagliardo–Nirenberg 插值不等式最佳常数，与Esteban和Loss用非线性流方法解决Caffarelli–Kohn–Nirenberg不等式最优对称区间的长期猜想，是不等式研究方向的奠基人之一。

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报告地点：同析4号楼308