The Schiffer conjecture asserted that if a bounded smooth domain 
with boundary homeomorphic
to the 
-sphere admits a nonconstant solution to the overdetermined
boundary value problem
 |
(1)
|
for some 
,
then 
is a ball.
Guo (2026) gave computer-assisted counterexamples consisting of bounded convex domains that are not balls and have
real-analytic spherical boundaries. The examples occur in dimensions
3, 4, 6, 8, 10, and 14. After rescaling, the constructions work for every 
. The existence proofs use computer-assisted contractions
in weighted polynomial coefficient spaces, with interval arithmetic on finite blocks and analytic
bounds on every infinite tail.
