Vladimir Tkachev's lecture notes
A collection of links to various PDE lecture notes
Text I: Fritz John. Partial Differential Equations, Springer-Verlag, New York, 1995.
Text II: Robert C. McOwen, Partial Differential Equations, Methods and Applications, Prentice Hall/Pearson Education, Inc., 2003 (Second Edition)
Occasionally: Lawrence C. Evans. Partial Differential Equations, AMS, Providence, RI.
Series: Graduate Studies in Mathematics, Vol. 19, 1998.
Here is an explanation of the method of reflection for the half-plane, section 4.2. .
Here is a sample final and solutions .
You can find the first day handout here .
Updated course contents, with sections from R. McOwen. some sections from F. John included (and more will be added).
Some extra useful reading: Notes on Cauchy-Kovalevskaya theorem and characteristic surfaces.
Notes on volumes, areas and masses
and an Ensglish-Swedish-French-German list of math terms by Christer Kiselman.
Things to know and remember for the
- Definitions and concepts: spaces, L^p, Sobolev, Banach,
L^p, L^infinity, Sobolev, etc, norms on Hilbert spaces; what is: a fundumental
solution; Green's function; bilinear form for a
second order differential operator; functional; linear
operator; adjoint of a differential operator; weak derivative;
convalution; Dirichlet problem and Neumann problem;
harmonic/subharmonic function; spherical mean and mean value
property; weak/strong maximum principle; Cauchy sequence and completeness; method of
images (refelction) for Green's function; Riesz representation theorem.
- Inequalities: Cauchy-Schwarz-Bunyakovsky, triangle, Minkovsky,
Holder, Poincare, Sobolev, mean value property for subharmonic functions;
- Formulas: second Green's identity, domain and single layer
potential, fundamental solution for Laplacian, Poisson integral formula/Poisson kernel;
- Embeddings: which L^p spaces are contained in which, Sobolev