## 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.