2nd lecture
On October 17, we covered the following:
- Harnack’s inequality for positive harmonic functions;
- A corollary: an increasing sequence of harmonic functions either converges almost uniformly to a (harmonic) function or is everywhere divergent to ;
- Green’s representation formula and the concept of Green’s function;
- Poisson kernel and the Poisson representation formula; solution of the Dirichlet problem in a ball;
- Corollaries of the Poisson representation formula: harmonic functions are real analytic, the mean value property characterizes harmonic functions etc.;
- Poincaré’s theorem: the Dirichlet problem in a smooth bounded domain has a unique smooth solution;
- The proof I presented is taken from Stefan Hildebrandt’s paper On Dirichlet’s principle and Poincaré’s méthode de balayage, Math. Nachr. 278 (no. 5), 2005; a variant can be found in Gilbarg and Trudinger (see chapter 2) or in my Polish lecture notes A short introduction to PDE.