Zero Set of Sobolev Functions with Negative Power of Integrability

is of Hausdorff dimension at most n− p, by a theorem of Federer-Ziemer [3]. And for functions of bounded variation, we have Hn−1 (Σ∗) = 0, see section 5.9 of [1] for more information on fine properties of BV functions. Hence our result concerning Σ, the Lebesgue set of u of the value zero makes sense because s > n− p. In order to show Theorem 1, we will need a Poincaré type inequality which will be proven in the next section.