SISSA  Universita` di Trieste
Corso di Laurea Magistrale in Matematica
A. A. 2010/2011
Prof. Ludwik Dąbrowski
Istituzioni di Fisica Matematica A
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 twodimensional space. Parabolic operators.
 General solution of a second order hyperbolic equation
with constant coefficients in the twodimensional space.
2. Wave equation.
 Vibrating string.
 Cauchy problem. D'Alembert formula.
 Some consequences of the D'Alembert formula.
 Semiinfinite 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.
 Illposedness 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. Oneparameter 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) AddisonWesley, 1983
W. Thirring. A course in mathematical physics vol. 3, Springer, 1981

Date esami 2011:
 3 Feb, 8:4512:30 scritto e 4 Feb orale
 24 Feb, 8:4512:30 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