Geometric Analysis

Prof. Antonio Greco
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The course is an introduction to the problem of determining the shape of solutions to boundary-value problems for second-order partial differential equations, mainly of elliptic type, occasionally parabolic.

Outline

– Review of the weak maximum principle, the strong maximum principle, and the Hopf lemma.
– Some motivations for Geometric Analysis and some characteristic results: the soap bubble theorem (Aleksandrov’s theorem), Serrin’s overdetermined problem, the Gidas-Ni-Nirenberg symmetry result.
– Convexity of solutions to the Dirichlet problem. Quasiconvexity.
– The Morse index of a solution and its role in Geometric Analysis. Work in progress.

Schedule

The course spans over 8 lectures of 2 hours each (16 hours total), one lecture per week. Details will be specified on the occasion of the first lecture, which will be given on October 20, 2023 at 4 p.m. in room B of the Department of Mathematics and Computer Science

Exam

The final exam consists in a presentation, and it can be recognized as 3.2 CFR.

References

1. Gidas, B.; Ni, Wei-Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209-243 (1979).
2. Berestycki, H.; Nirenberg, L. On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat., Nova Sér. 22, No. 1, 1-37 (1991).
3. Fraenkel, L. E. An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press. x, 340 p. (2011).
4. Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. Prentice-Hall, Inc. X, 261 p. (1967).
5. Serrin, James. A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304-318 (1971).
6. Sperb, Rene P. Maximum principles and their applications. Academic Press. IX, 224 p. (1981).