Variational Methods for Partial Differential Equations
Ph.D. Program in Mathematics / Ph.D. Program in Mathematics and Computer Science
University of Messina
30 hours
This doctoral course focuses on variational methods for differential equations, with particular emphasis on differentiable functionals on Banach spaces, Sobolev spaces, critical point theory, nonlinear elliptic problems, Kirchhoff-type equations, and fractional Laplacian problems.
This course was offered in two editions: in 2006 within the Ph.D. Program in Mathematics and in 2017 within the Ph.D. Program in Mathematics and Computer Science. Section 5 was introduced only in the 2017 edition, while Section 6 was taught by Prof. Luca Vilasi as a 6-hour module.
Course Contents
1. Gâteaux and Fréchet Differentiable Functionals on Banach Spaces
Introduction to variational methods. 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.
2. Sobolev Spaces
The Sobolev space W1,p(]a,b[). Completeness, reflexivity and separability of Sobolev spaces. Hölder spaces. The space W01,p(]a,b[). Poincaré inequality. Sobolev embedding theorems. The Sobolev spaces W1,p(Ω) and W01,p(Ω): completeness, reflexivity, separability and embedding theorems. Differentiability of the norm. Variable exponent Sobolev spaces W1,p(x)(Ω) and W01,p(x)(Ω). The modular of the space and its properties.
3. Variational Methods: A First Example
Study of a second-order ordinary boundary value problem. Functionals associated with the problem. Minimization of coercive functionals. Existence theorem for the boundary value problem.
4. A General Variational Principle and Applications
Parameter-dependent minimization problems. Functionals of the form Φ + λΨ. Ricceri's 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.
5. Variational Methods for a Kirchhoff-Type Problem
Ricceri's three critical points theorem for functionals of a suitable class. Properties of the p(x)-Laplacian operator. Two existence results providing three solutions for a Kirchhoff-type problem involving the p(x)-Laplacian.
6. Variational Methods for a Problem Involving the Fractional Laplacian
Fractional Sobolev spaces Ws,p. Gagliardo's definition and the main properties. Embedding theorems. The case p = 2 and its connection with the fractional Laplacian. Definition of Ws,2 by means of the Fourier transform and its equivalence with Gagliardo's definition. The fractional Laplacian operator. Equivalence among different definitions of the fractional Laplacian. Study of a problem involving the fractional Laplacian by variational methods: direct minimization approach and the Mountain Pass Theorem.
References
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer.
- Z. Denkowski, S. Migórski, N. S. Papageorgiou, An Introduction to Nonlinear Analysis, Springer, New York, 2003.
- B. Ricceri, A General Variational Principle and Some of Its Applications, Journal of Computational and Applied Mathematics 113 (2000), 401–410.
- B. Ricceri, A Further Three Critical Points Theorem, Nonlinear Analysis 71 (9), 4151–4157, 2009.
- 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.
- F. Cammaroto, L. Vilasi, Multiple Solutions for a Kirchhoff-Type Problem Involving the p(x)-Laplacian Operator, Nonlinear Analysis 74, 1841–1852, 2011.
- F. Cammaroto, L. Vilasi, Weak Solutions for a Fractional Equation in Rn with Kirchhoff Terms, Complex Variables and Elliptic Equations 61 (10), 1362–1374, 2016.