Textbooks:

• 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 final .

• Definitions and concepts: spaces, L^p, Sobolev, Banach, Hilbert, etc; norms, 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 embedding theorems.