November 16th, 2020

November 18th, 2020

November 20th, 2020

November 17th, 2020

November 19th, 2020

Lecture's slides

Mon.-Fri. Lectures 10.00-13.00

The topic of this RISM course will be the validity, at different levels of generality, of the maximum Principle for elliptic degenerate operators and possible applications.

The course will include the following main topics:

  1. Elementary cases: linear, convex and harmonic funtions.
  2. Maximum Principle for uniformly elliptic operators on bounded domains. The theorem Alexandrov-Bakelman-Pucci (ABP). Applications and a priori estimates.
  3. Generalized subharmonic funtions, super-solutions in the sense of Calabi, viscosity supersolutions. Nonlinear version of th ABP theorem.
  4. Maximum Principle for fully nonlinear operators on bounded domains.
  5. Discrete Perron-Frobenius’ theory. Collatz-Wielandt representation formulas. On the validity of the Maximum Principle and positivity of the principal eigenvalue.


  1. M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, 1967
  2. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983
  3. L. Hormander, Notions of Convexity, Modern Birkhauser Classics, Birkhauser Basel, 1994
  4. L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Colloquium Publications, Vo. 43, AMS 1995
  5. M. G. Crandall, H. Ishii and P. L. Lions, User's Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bulletin Ams, Vol. 27, No. 1, 1992
  6. M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkhauser Classics, Birkhauser Basel, 2008
  7. H. Berestycki, L. Nirenberg, and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47 (1994), no. 1
  8. I. Capuzzo-Dolcetta, F. Leoni, A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Communications in Partial Differential Equations 30 (12), 1863-1881, 2005
  9. I. Capuzzo Dolcetta, A. Vitolo, Directional ellipticity on special domains: weak Maximum and Phragmen-Lindelof principles,  Nonlinear Analysis 184, 69-82, 2019
  10. H. Berestycki, I. Capuzzo Dolcetta, A. Porretta, L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, Journal de Mathématiques Pures et Appliquées 103 (5), 1276-1293, 2015

Villa Toeplitz

Via G.B. Vico, 46
21100 Varese VA

Daniele Cassani
Daniele CassaniRISM Director

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