Affiliation:
1. Department of Mathematics, University of Chicago , Chicago , IL 60637 , USA
Abstract
Abstract
Let
u
u
be a nontrivial harmonic function in a domain
D
⊂
R
d
D\subset {{\mathbb{R}}}^{d}
, which vanishes on an open set of the boundary. In a recent article, we showed that if
D
D
is a
C
1
{C}^{1}
-Dini domain, then, within the open set, the singular set of
u
u
, defined as
{
X
∈
D
¯
:
u
(
X
)
=
0
=
∣
∇
u
(
X
)
∣
}
\left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\}
, has finite
(
d
−
2
)
\left(d-2)
-dimensional Hausdorff measure. In this article, we show that the assumption of
C
1
{C}^{1}
-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite
ℋ
d
−
2
{{\mathcal{ {\mathcal H} }}}^{d-2}
-measures.
Subject
General Mathematics,Statistical and Nonlinear Physics