Qualitative Properties of Solutions of Uniformly Parabolic Equations
Prof. Daniele Castorina
Università di Napoli Federico II
Dott. Simone Ciani
Università di Bologna
Abstract
The course is divided in two sections:
Section 1 – Second Order Parabolic Equations (Ciani)
In the first lectures we will follow Ch. VII of [1], by setting up a definition of solution of parabolic uniformly elliptic equations. In the sequel we will prove existence and uniqueness for the Cauchy-Dirichlet problem, thanks to the method of Galerkin approximations and a priori estimates. Then we will approach regularity theory: first we will prove that the unique solution to the boundary value problem proposed improves its regularity as much as the initial value datum allows, until we reach the smoothness C-infinity by a bootstrap argument. In the last lecture, time permitting, we will comment on the lack of regularity when the coefficients and the data are rough; and give a glimpse of the possible minimal regularity properties affordable, following Chap XI-XII of [2].
Section 2 – The Alexandrov-Bakelman-Pucci method and its applications (Castorina)
The classical Alexandrov-Bakelman-Pucci (or ABP) estimate is a uniform bound for strong solutions of second order uniformly parabolic operators with bounded measurable coefficients written in nondivergence form. Its main feature is being a basic tool in the regularity theory for fully nonlinear parabolic equations. However, the ABP method is fairly general and it can be adapted to a wide variety of different issues such as obtaining a maximum principle in domains of small measure, as well as simplifying the proofs of several isoperimetric and Sobolev inequalitiees. The aim of this second part course is to introduce the ABP method in detail, discuss some of its generalizations and refinements and to give a detailed and complete overview of its applications, explicitly highlighting the improvements of using this technique with respect to previous and more classical tools.
Outline
- Existence and Uniqueness
- Regularity Theory I – Improvement of Regularity
- Minimal Regularity II – Hölder Continuity, Harnack inequality and Applications
- The Alexandrov-Bakelman-Pucci estimate
- The Maximum Principle in small domains
- The Gidas-Ni-Nirenberg Theorem and Isoperimetric and Sobolev inequalities
Schedule
- 8/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
- 9/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
- 10/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
Exam
The final assessment consists of one of the two choices: a list of exercises to solve (during the course) and a seminar; or a written elaborate on selected topics of the course.
References
- L. Evans, Partial Differential Equations, Second Edition, AMS, 1998.
- E. DiBenedetto, U. Gianazza, Partial Differential Equations, Third Edition, Birkhäuser, 2023.
- Berestycki, H., Nirenberg, L. On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22, 1991, 1–37.
- Berestycki, H., Nirenberg, L., Varadhan, S. R. S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47,1994, 47–92.
- Cabré, X. On the Alexandrov-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48, 1995, 539–570.
- Cabré, X., Ros-Oton, X. Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations 255, 2013, 4312–4336.
- Cabré, X., Ros-Oton, X., Serra, J. Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. 18, 2016, 2971–2998.
- Gilbarg, D., Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin-New York, 1983.