Best oscillation estimate for rectangular membrane strips

Determine the best upper bound for the length of an interval on which a solution of the vibrating membrane equation on a rectangle can remain nonnegative in a horizontal strip, expressed as a function of the strip endpoints α and β.

Background

For a rectangular membrane and a horizontal strip, Proposition 7 gives the general estimate that a nonzero solution cannot remain nonnegative on the strip for an interval of length at least 2π. A stronger estimate is available when the strip spans the full horizontal width, but the authors explicitly state that they do not know the best general estimate depending on the strip endpoints. The unresolved issue concerns the sharp dependence of the oscillation-time bound on the geometry of the strip.

References

We have no idea of what could be the best estimate in general as a function of $(\alpha, \beta)$.

— Uniform oscillation properties of almost periodic functions  (2610.06515 - Haraux, 5 Oct 2026) in Remark following Proposition 7 (Section 7, “Back to vibrating membranes”)