## Introduction to Mathematical Physics

### Prof. Boris Dubrovin

Contents:

- Fundamental equations in Mathematical Physics. Oscillator, Helmholtz, heat, Laplace and Poisson equations. Classification of second order partial differential equations
- Boundary problems for linear second order partial differential equations. The Cauchy problem. Cauchy-Kowalevskaya theorem. Boundary problem for elliptic equations. Well-posed problems.
- Banach and Hilbert spaces. L2 and C spaces. Selfadjoint operators.
- Separation of variables for wave, Laplace and Poisson equations
- The spectrum of the Sturm-Liouville operator. Particular cases and classic orthogonal polynomials (connection with random matrices). Spherical functions.
- Elements of nonlinear equations theory. Hopf and Kortweg-de Vries equations.

Bibliography:

- A. Tikhonov, A. Samarskij, Equazioni della fisica matematica (Edizioni MIR, 1977)
- R. Courant, D. Hilbert, Methods of Mathematical Physics (New York Intersci. Publ., 1989)