Differential Geometry Seminar: On some geometric aspects of partial differential equations: soap bubbles, overdetermined problems, and related questions

Speaker: Giorgio Poggesi (University of Adelaide)

Title: On some geometric aspects of partial differential equations: soap bubbles, overdetermined problems, and related questions

Date: Friday 16 May 2025, 12:10pm in Lower Napier LG28 Lecture Theatre

Abstract: The talk will focus on some geometric aspects of partial differential equations (PDEs). We will discuss various topics that share the common feature of showing an interplay between analysis and geometry. In particular, we will review the celebrated symmetry theorems established by Alexandrov in 1958, Serrin in 1971, and Gidas, Ni, and Nirenberg in 1979. The first – Alexandrov’s soap bubble theorem – deals with constant mean curvature hypersurfaces. The second – Serrin’s symmetry result –  concerns certain overdetermined problems for PDEs. The last – the Gidas-Ni-Nirenberg Theorem – addresses the question of whether solutions to boundary value problems for PDEs inherit symmetry properties from the domain. Each of these theorems inaugurated incredibly fruitful research directions that continue to play distinguished roles in the contemporary research.

We will present various recent generalisations of these celebrated symmetry theorems and discuss related quantitative stability results.