## Linear partial differential equations of mathematical physics (Equazioni differenziali alle derivate parziali)

• Programma (eng):

1. Linear partial differential operators.
- Definitions and main examples.
- Principal symbol of a linear differential operator.
- Change of independent variables.
- Canonical form of linear differential operators of order 1 and of order 2, with constant coeffcients.
- Characteristics. Elliptic and hyperbolic operators.
- Reduction to a canonical form of second order linear differential operators in a two-dimensional space. Parabolic operators.
- General solution of a second order hyperbolic equation with constant coefficients in the two-dimensional space.

2. Wave equation.
- Vibrating string.
- Cauchy problem. D'Alembert formula.
- Some consequences of the D'Alembert formula.
- Semi-infinite vibrating string.
- Periodic problem for wave equation.
- Introduction to Fourier series.
- Finite vibrating string. Standing waves.
- Energy of vibrating string.
- Solutions in dimension 2 and 3.
- Solutions of the inhomogeneous problem.

3. Laplace equation.
- Ill-posedness of Cauchy problem for Laplace equation.
- Dirichlet and Neumann problems for Laplace equation on the plane.
- Properties of harmonic functions: mean value theorem, the maximum principle.
- Harmonic functions on the plane and complex analysis.

4. Heat equation.
- Derivation of heat equation.
- Main boundary value problems for heat equation.
- Fourier transform.
- Solution of the Cauchy problem for the heat equation on the line.
- Mixed boundary value problems for the heat equation.
- More general boundary conditions.
- Solution of the inhomogeneous heat equation.

5. Abstract Cauchy problem. One-parameter evolution semigroups.
- Notes on Schroedinger equation.
- Notes on Maxwell equation.
- Notes on Dirac equation.
• Bibliografia:

Lecture Notes by Boris Dubrovin

L.C. Evans, Partial differential equations, Providence, AMS, 1998

H.O. Fattorini, The Cauchy problem (Enc. of Math and Appl. vol 18) Addison-Wesley, 1983

W. Thirring. A course in mathematical physics vol. 3, Springer, 1981
• Date esami 2011:
- 3 Feb, 8:45-12:30 scritto e 4 Feb orale
- 24 Feb, 8:45-12:30 scritto e 25 Feb orale
- 16 Giu, 9:30 - 12:30 scritto e dalle 15:00 orale
- 21 Lug, 9:30 - 12:30 scritto e dalle 15:00 orale
- 9 Sett, 9:30 - 12:30 scritto e dalle 15:00 orale
- 28 Sett, 9:30 - 12:30 scritto e 29 Sett orale