A Computer-Assisted Counterexample to a Sharp L<sup>p</sup>-Dissipativity Conjecture for the Three-Dimensional Lam<span class="">é Operator

Source abstract

Maz'ya's Problem 43 asks whether the sharp planar condition for Lp-dissipativity of the Lamé operator remains sufficient in dimension three. We give a negative answer by an exact computer-assisted construction. At Poisson ratio $\nu=0$ and $p=18+12\sqrt2$, the proposed condition holds with equality. We exhibit an explicit nonvanishing periodic vector field and certify, through a finite Fourier dual certificate evaluated in exact rational arithmetic, that the associated transformed form has strictly negative mean. A quantitative localization argument converts this periodic obstruction into a compactly supported smooth test field in every nonempty open subset of $R^3$. The certificate, two verification scripts, a readable coefficient export, and cryptographic checksums accompany the source package.