 
Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 17, Number 2, Pages 243265
S 08940347(04)004539
Article electronically published on February 11, 2004
ON NEUMANN EIGENFUNCTIONS IN LIP DOMAINS
RAMI ATAR AND KRZYSZTOF BURDZY
1. Introduction and main result
A planar set D will be called a lip domain if it is Lipschitz, open, bounded,
connected, and given by
(1) D = {(x1, x2) : f1(x1) < x2 < f2(x1)},
where f1, f2 are Lipschitz functions with constant 1. The assumption that D is a
Lipschitz domain puts an extra constraint on the functions fk; we discuss this issue
in greater detail later in this section.
Let µ2 denote the second eigenvalue for the Laplacian in D with Neumann bound
ary conditions. Here is our main result.
Theorem 1. (i) The second eigenvalue µ2 is simple in all lip domains except
squares.
(ii) ("Hot spots conjecture" ) For every lip domain, every Neumann eigenfunction
corresponding to µ2 attains its maximum and minimum at boundary points only.
