21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

Hirata, Kentaro : The 10th COE Lecture Series

Martin's representation for positive harmonic functions and the existence of positive solutions of nonlinear elliptic equations

It is well known that every positive harmonic function in a ball can be represented as Poisson's integral. In 1941, R. S. Martin obtained such representations on general domains by introducing an ideal boundary (Martin boundary) and an integral kernel (Martin kernel). The question is for what domains are the Martin boundaries homeomorphic to the Euclidean boundaries? The boundary Harnack principle is one of useful tools for this. Also, it gives useful estimates for the Green function to study the existence of positive solutions of nonlinear elliptic equations.

The main aims of this talk are
1. to show the boundary Harnack principle in a uniform
domain, and then study the Martin boundary of it;
2. to show the existence of positive solutions, with the
same growth as the Martin kernel at infinity, of
certain nonlinear elliptic equations in a uniform cone.

The plan of this talk is as follows. In the first day, we will state elementary properties of harmonic and subharmonic functions. Also, we will sketch out Martin's representation of positive harmonic functions in general domains.
In the second day, we will show that the Martin boundary of a uniform domain is homeomorphic to the Euclidean boundary, establishing the boundary Harnack principle.
In the final day, we will show the existence of positive solutions of certain nonlinear elliptic equations in a uniform cone.

21st Century COE Program : Mathematics of Nonlinear Structure via Singularity