Lectures in analysis for Ph.d-students

The proof is exceedingly innovative and relies upon a closure theorem for Dirichlet series. Lecture 2 deals with a variational problem for product measures which is used to solve a non-linear Cauchy problem introduced by Schrödinger to describe phenomena in quantum physics. Lecture 3 gives an exposition about Beurling’s studies of free boundary problems for the Laplace operator related to Helmholtz’ equations. We have included § 3.B which contains the essential parts in Beurlings conformal mapping theorem from his article [Beurling:xx: Acta 1953]. Sections 4-6 expose results by Carleman in PDE-theory. The material in § 4 is classic but a notable point occurs after Theorem 1.2 in § 4 which in addition to standard asymptotic formulas for evaluations of eigenfunctions also give asymptotiic formulas when one passes to higher order derivatives. In § 6 one studies the following non-linear boundary value problem: A C-domain Ω is given in R and F (u, p) is a non-negative continuous function on R × ∂Ω is such that u 7→ F (u, p) is strictly increasing for each fixed p ∈ ∂Ω and tends uniformly to +∞ as u → +∞. Carleman proved that for every pair Ω, F as above and each p∗ ∈ Ω there exists a unique function u which is harmonic except for the Newtonian singularity at p∗ and the inner normal ∂u ∂n satisfies ∂u ∂n (p) = F (u(p), p) : p ∈ ∂Ω The proof employs a homotopy method where non-linearity increases in a parametrized system of PDE-equations. § 5 describes to Carleman’s construction of fundamental solutions to elliptic