Variational Principles and Applications
Ph.D. Program in Mathematics
University of Messina
30 hours
This doctoral course focuses on variational principles and their applications to nonlinear analysis, with particular emphasis on Ekeland's Variational Principle, Gâteaux and Fréchet differentiable functionals on Banach spaces, critical point results, and applications to nonlinear differential problems.
Course Contents
1. Ekeland's Variational Principle
Review of the theory of complete metric spaces. Lower semicontinuous functions and their characterizations. Ekeland's Variational Principle: statement, proof and geometric interpretation. First applications to nonlinear analysis. Caristi's Fixed Point Theorem.
2. Gâteaux and Fréchet Differentiable Functionals on Banach Spaces
Gâteaux differentiable functionals defined on Banach spaces. Gâteaux derivative and its fundamental properties. Fréchet differentiable functionals defined on Banach spaces. Fréchet derivative and its fundamental properties. Comparison between the two notions of differentiability. Chain rule for differentiable functionals. Mean Value Theorem for Fréchet differentiable functionals. Product rule for differentiable functionals. Second-order Gâteaux and Fréchet derivatives. Necessary conditions for minima and existence results. Functionals associated with differential problems. Applications of Ekeland's Variational Principle to Gâteaux differentiable functionals.
3. A Complement to Ekeland's Variational Principle
A complement to Ekeland's Variational Principle. Applications to Caristi's Fixed Point Theorem and to Gâteaux differentiable functionals. A Liouville-type theorem for harmonic functions on exterior domains. A Liouville-type theorem for the homogeneous wave equation. Extensions and possible developments.
4. A General Variational Principle and Applications
Parameter-dependent minimization problems. Functionals of the form Φ + λΨ. A general variational principle. Conditions ensuring the existence of critical points and multiplicity results. Some applications to nonlinear elliptic problems. Existence of infinitely many solutions for a two-point boundary value problem with homogeneous Dirichlet boundary conditions. Existence of infinitely many solutions for the Dirichlet problem involving the p-Laplacian.
References
- I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, SIAM, 1987.
- D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Bombay, 1989.
- B. Ricceri, A General Variational Principle and Some of Its Applications, Journal of Computational and Applied Mathematics 113 (2000), 401–410.
- Z. Denkowski, S. Migórski, N. S. Papageorgiou, An Introduction to Nonlinear Analysis, Springer, New York, 2003.
- F. Cammaroto, A. Chinnì, A Complement to Ekeland's Variational Principle in Banach Spaces, Bulletin of the Polish Academy of Sciences. Mathematics 44 (1), 29–33, 1996.
- F. Cammaroto, A. Chinnì, A Liouville-Type Theorem for Harmonic Functions on Exterior Domains, Journal of Mathematical Analysis and Applications 244, 1–9, 2000.
- F. Cammaroto, A. Chinnì, A Liouville-Type Theorem for the Homogeneous Wave Equation, Le Matematiche LVII (I), 167–170, 2002.
- F. Cammaroto, A. Chinnì, Infinitely Many Solutions for a Two-Point Boundary Value Problem, Far East Journal of Mathematical Sciences 11 (1), 41–51, 2003.
- F. Cammaroto, A. Chinnì, B. Di Bella, Infinitely Many Solutions for the Dirichlet Problem Involving the p-Laplacian, Nonlinear Analysis 61, 41–49, 2005.