Teacher: Prof. Antonio Greco
Duration: 8 lectures of 2 hours each (16 hours total)
Timetable: 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
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.
1. Review of the weak maximum principle, the strong maximum principle, and the Hopf lemma.
2. 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.
3. Convexity of solutions to the Dirichlet problem. Quasiconvexity.
4. The Morse index of a solution and its role in Geometric Analysis. Work in progress.
Gidas, B.; Ni, Wei-Ming; Nirenberg, L.
Symmetry and related properties via the maximum principle.
Commun. Math. Phys. 68, 209-243 (1979).
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).
Fraenkel, L. E.
An introduction to maximum principles and symmetry in elliptic problems.
Cambridge University Press. x, 340 p. (2011).
Protter, Murray H.; Weinberger, Hans F.
Maximum principles in differential equations.
Prentice-Hall, Inc. X, 261 p. (1967).
A symmetry problem in potential theory.
Arch. Ration. Mech. Anal. 43, 304-318 (1971).
Sperb, Rene P.
Maximum principles and their applications.
Academic Press. IX, 224 p. (1981).
Credits: 3,2 CFR (with final talk)