A Counterexample to the One-Set Sufficiency Suggested in Maz'ya's Problem 56
Abstract
Let $n\ge 3$, let $\Omega\subset\R^n$ be a bounded smooth domain, and put $Q=\Omega\times(0,T)$. We construct a finite nonnegative measure $\mu$, absolutely continuous with respect to Lebesgue measure and supported in a fixed compact subcylinder of $Q$, such that \[ \mu(K)\le C\,\capheatQ(K) \] for every compact $K\subset Q$, where $\capheatQ$ is Pierre's classical heat-equation capacity. With the explicit half-closed-cylinder normalization introduced below, the same estimate holds for every compact $K\subset\Omega\times[0,T)$. Nevertheless, the first trace inequality in Maz'ya's Problem~56, \[ \int_Q |u|^2\,\dd\mu \le C\left(\int_Q |\nabla_xu|^2\,\dd x\dd t +\esssup_{0<t<T}\|u(t)\|_{L^2(\Omega)}^2\right), \] fails even for $u\in C_c^\infty(Q)$. We state the exact $p=2$ variational capacity used in the proof, identify it with Pierre's capacity under the hypotheses of the construction, and distinguish the open-cylinder and half-closed-cylinder conventions. We also prove that a natural strong multilevel capacitary estimate would imply the trace inequality, and then exhibit nested compact sets and levels for which that estimate fails. Finally, we quantify why the same test sequence has divergent $L^2(0,T;H^{-1}(\Omega))$ time-derivative energy. Thus the one-set sufficiency suggested in Problem~56 is false for the first trace inequality in every dimension $n\ge3$. The second inequality in that problem and the cases $n=1,2$ are not resolved here.
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