报告题目：From Classification of solutions to Quantitative Stability in Nonlinear Elliptic PDEs

报告摘要：Classification and rigidity results are a classical topic in the study of nonlinear elliptic partial differential equations. They identify family of solutions enjoying a prescribed set of qualitative properties and often reveal deep connections between PDEs, functional inequalities, and geometric problems. A related and challenging question is whether such rigidity persists in a quantitative form: roughly speaking, whether or not almost solutions to the equation must be close to the corresponding model solutions. This mini-course will present both classical results and some recent developments in this direction, with particular emphasis on classification and quantitative stability for nonlinear elliptic equations. We will discuss the role of symmetry methods, maximum principles, integral identities, and illustrate how these tools lead both to classification theorems and to quantitative estimates. Topics will include critical (p)-Laplace equations and their anisotropic counterparts, Liouville-type equations, semilinear elliptic equations and quantitative versions of the Gidas–Ni–Nirenberg symmetry theorem, as well as stability results related to Sobolev inequalities and overdetermined problems. We will also discuss how these ideas extend to non-Euclidean settings, including elliptic equations on Riemannian manifolds, and how PDE rigidity can be connected to geometric rigidity. The main goal is to present a unified point of view on a recurring phenomenon in nonlinear analysis: exact rigidity at the qualitative level and its quantitative stability counterpart.

Lesson 1-Maximum principles for linear elliptic PDEs: weak and strong maximum principle,Hopf Lemma, maximum principle for thin domains, first eigenfunction. ABP method and maximum principle for small domains.

Lesson 2-The method of moving planes (MMP).A simple application: the parallel surface problem.Introduction to quantitative studies by the MMP.

Lesson 3-Alexandrov Soap Bubble Theorem and Serrin’s overdetermined problem.Proofs by moving planes and integral identities(P-function).Remarks on their quantitative counterparts.

Lesson 4-Gidas-Ni-Nirenberg Theorem in a ball: proof by MMP. Symmetry for the critical equation: an integral formulation of the MMP.

Lesson 5-Caffarlli-Gidas-Spruck Theorem: a new approach by using the P-function and application to the classification of Riemannian manifolds with nonnegative Ricci curvature.

报告人简介：Giulio Ciraolo，意大利米兰大学数学系（Federigo Enriques）数学分析正教授，定量几何分析方向中生代代表学者，发表论文60余篇。师从Rolando Magnanini教授，博士后导师为Figalli（Fields奖得主）。在J.reine angew.Math.、Int.Math.Res.Not.、Arch.Ration.Mech.Anal.等国际顶尖期刊发表论文。

报告时间：

报告地点：同析4号楼308教室