Title: Minimum principles and positive invertibility of linear second order elliptic operators.
Contents:
- Classical minimum principles revisited.
- Generalized minimum principle of Protter and Weinberger.
- Uniform decaying property of Walter.
- Classification theorems of supersolutions.
- The existence of the principal eigenvalue.
- Characterization of the positive invertibility. Theorem of Amann, López-Gómez and Molina-Meyer.
- Some consequences.
Abstract:
In this course, we will study the classical minimum principles of E. Hopf [1, 2], M. H. Protter and H. F. Weinberger [5], and the more recent theorems on the classification of supersolutions by W. Walter [6] and J. López-Gómez [3]. As a result of these classification theorems, the generalized minimum principle of M. H. Protter and H. F. Weinberger [1] is complemented and substantially refined. Finally, we discuss some important consequences of these results. This course is based on the book [4]. References:
- Hopf, E. (1927). Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsber. Preuss. Akad. Wiss. 19, pp. 147–152.
- Hopf, E. (1952). A remark on linear elliptic differential equations of the second order, Proc. of the Amer. Math. Soc. 3, pp. 791–793.
- López-Gómez, J. (2003). Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns. 8, pp. 1025–1042.
- López-Gómez, J. (2013). Linear second order elliptic operators (Hackensack, NJ: World Scientific).
- Protter, M. H. and Weinberger, H. F. (1967). Maximum Principles in Differential Equations (Prentice-Hall, Englewood Cliffs).
- Walter, W. (1989). A theorem on elliptic differential inequalities and applications to gradient bounds, Math. Z. 200, pp. 293–299.
Date:
August 26-30, 2024 Time:
- 10:00 - 11:20 (August 26-29);
- 09:30 - 12:10 (August 30)
Local: Soon
Lecturer: Julián López-Gómez
Bibliography:
The professor Julián Lómez-Gómez was the advisor of 15 PhD Thesis, author of more than 220 research papers, most of them published in leading journals, 10 books, and editor of 6 special volumes, among them, the Proceedings of the 10th AIMS Conference (Madrid 2014), the largest congress on Differential Equations ever held with over 2300 participants. He has delivered invited talks in 87 conferences, and organized, or co-organized, 32 scientific meetings, served as member of the editorial boards of Advanced Nonlinear Studies, Nonlinear Analysis, Theory, Methods and Applications, Discrete and Continuous Dynamical Systems, Rendiconti dell’Istituto di Matematica dell’Universt`a di Trieste (RIMUT), Abstract Applied Analysis, Journal of Applied Mathematics, Applied Mathematics, and has elaborated reports for 80 interdisciplinary journals and National Evaluation Agencies including Journal of Differential Equations, Nonlinear Analysis TMA, Nonlinear Analysis RWA, Advanced Nonlinear Studies, Archive of Rational Mechanics and Analysis, Transactions of the American Mathematical Society, Zeitschrift f¨ur angewandte Mathematik und Physik, Oykos and Web Ecology. Mainly recognized for his contributions in the following fields:
- Bifurcation and Operator Theory: characterization of nonlinear eigenvalues for Fredholm operators yielding to the unique algebraic multiplicity that detects any change of the topological degree.
- Theory of Partial Differential Equations: including the characterization of the strong maximum principle for elliptic and periodic-parabolic operators, the invertibility of 1-D non-cooperative systems, the theory of metasolutions, as well as a number of ground-breaking contributions to the theory of superlinear indefinite problems.
- Applications of PDEs in Biology and Ecology: including the failure of the principle of competitive exclusion in the presence of refuges and the most pioneering results on permanence through segregation.
