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 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