Differential Equations 1
Academic year 2012/13 Lecturer: Annalisa Cesaroni
Dipartimento di Matematica, office 519 Torre Archimede, tel. 049 827 1442
email [email protected]
web page http://www.math.unipd.it/ acesar/
Classes :Monday, Tuesday 9.30-11.15, room 2AB45.
Program The main theme of the course will be an introduction to linear partial differential equations.
Plan:
- Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem.
- Weak and strong maximum principle for elliptic operators.
- Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity.
- Wave equation, existence of the solution, D’Alembert formula, solutions by spher- ical means, main properties, uniqueness by energy methods.
- The Method of characteristics for first order equations, linear and nonlinear, trans- port equation, Hamilton-Jacobi equation, scalar conservation laws.
Bibliography
• L.Evans Partial Differential Equations, AMS 2010 (2nd edition)
• D. Gilbarg, N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 1998.
• E. Di Benedetto Partial Differential Equations, Birkauser, 2010 (2nd edition).
• W. A. Strauss Partial Differential Equations. An Introduction, Wiley, 1992.
Prerequisites
Differential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory.
Exam Final written exam, with both exercises and theoretic questions, about definitions, statements of theorems, (sufficiently short) proofs or part of long proofs.
Every 2 or 3 weeks there will be given homeworks (I will put it on the web page on Tuesday, and they will be due for the next Monday) that will be used to compute the final grade (up to 4 points more)
Students may accept the grade obtained in this way or may ask to take an oral exam.
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