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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 17, Number 2, Pages 243265
S 0894-0347(04)00453-9
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 2 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 2 is simple in all lip domains except
squares.
(ii) ("Hot spots conjecture" ) For every lip domain, every Neumann eigenfunction
corresponding to 2 attains its maximum and minimum at boundary points only.

  

Source: Atar, Rami - Department of Electrical Engineering, Technion, Israel Institute of Technology

 

Collections: Engineering