Rayleigh's clamped-plate conjecture states that, among open sets of a given finite positive volume in 
, the ball minimizes the
first eigenvalue of the biharmonic
operator with clamped boundary conditions. On a smooth boundary,
these conditions require both the eigenfunction and its normal derivative to vanish.
Nadirashvili (1995) proved the planar case. Ashbaugh and Benguria (1995) proved the
result in dimensions 2 and 3.
Suragan (2026, Theorem 1.1) reports the result in dimensions 4 and 5 for arbitrary open sets of finite positive volume.
In these dimensions equality holds precisely for a translated ball
with a relatively closed set of zero 
Sobolev capacity removed. The argument compares the first
clamped-plate eigenvalue with the square of the first
Dirichlet eigenvalue, with a sharp ratio
attained on balls.
