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Summary: THE FUNDAMENTAL GAP CONJECTURE FOR POLYGONAL
DOMAINS
ZHIQIN LU AND JULIE ROWLETT
Abstract. In 1985, S. T. Yau made the following "fundamental gap conjec-
ture," [25]. For a convex domain Rn,
(0.1) () := d2
(2() - 1()) 32
where d is the diameter of the domain, and 0 < 1() < 2() are the
first two eigenvalues of the Euclidean Laplacian on with Dirichlet boundary
condition. The scalar invariant is the gap function. We restrict attention to
planar domains. Our main result is a compactness theorem for the gap function
when the domain is a triangle in R2. This result shows that for any triangles
which collapse to the unit interval, the gap function is unbounded. Due to
numerical methods, we expect that the fundamental gap conjecture holds for
all triangular domains in R2. We show with examples that the behavior of
the gap for collapsing polygonal domains is quite delicate. These examples
motivate a technical result for collapsing polygonal domains giving conditions
under which the gap function either remains bounded or becomes infinite.
Our work initiates a general program to prove the fundamental gap conjecture
using convex polygonal domains.
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