TCC course: Criticality theory for Schrödinger operators

Prof. Vitaly Moroz (Swansea University)

Thursdays 2pm - 4pm on MS Teams, starting on 20th January and finishing on 10th March 2022

This course is run as part of the Taught Course Centre organised in collaboration with Bristol, Imperial, Oxford and Warwick.

This course is an introduction to Agmon's Criticality Theory of Schrödinger operators. It will focus on two core but not widely known ideas, namely Allegretto-Piepenbrink positivity principle and Phragmen-Lindelöf comparison principle. We will see how these fundamental principles enable to prove a range of Hardy type inequalities, and at the same time provide a powerful tool in the analysis of the structure of positive solutions for large classes of nonlinear elliptic equations. The course splits into two parts:

Linear theory:

1. Allegretto-Piepenbrink positivity principle for linear Schrödinger operators and some corollaries
2. Connection with Hardy inequalities
3. Maximum principle on bounded and ubounded domains
4. Phragmen-Lindelöf comparison principle: large and small positive solutions
5. Weak, strong and critical potentials

Nonlinear applications:

1. nonlinear Liouville theorems, Serrin's critical exponent, fast and slow decay solutions
2. singular solutions of semilinear elliptic equations, local Keller-Osserman bound, removable singularities
3. boundary blow-up solutions of semilinear elliptic equations in bounded domains, global Keller-Osserman bound

The course prerequisites are limited to basic concepts of elliptic PDEs: weak solutions, classical maximum principle, basic understanding of Sobolev spaces.

Handwritten notes:

• Notes Week 1.pdf
• Notes Week 2.pdf
• Notes Week 3.pdf
• Notes Week 4.pdf
• Notes Week 5.pdf
• Notes Week 6.pdf
• Notes Week 7.pdf
• Notes Week 8.pdf

Homework:

• Exercises Week 1.pdf
• Exercises Week 2.pdf
• Exercises Week 3-4.pdf
• Exercises Week 5-6.pdf
• Exercises Week 7-8.pdf

Bibliography:

• D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983.
• M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, 1984.
• S. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, Methods of functional analysis and theory of elliptic equations (Naples, 1982), Liguori, Naples, 1983, pp.19-52.
• V. Moroz, Positivity principles and decay of solutions in semilinear elliptic problems, (Lecture Notes, Padova 14-17 September 2020). DOI:10.13140/RG.2.2.10445.05601
• M. Marcus, V. Mizel, Y. Pinchover, On the best constant for Hardy's inequality in R^n. Trans. Amer. Math. Soc. 350 (1998), 3237-3255.
• C. Bandle, V. Moroz, W. Reichel, `Boundary blowup' type sub-solutions to semilinear elliptic equations with Hardy potential. J. Lond. Math. Soc. (2) 77 (2008), 503-523.