Isocapacity inequalities in the theory of Sobolev Spaces

Question

Section 9 of the lectures notes of Maz'ya (Мазья) on isocapacity, lectures notes that can be found at: http://www.math.liu.se/~vlmaz/pdf/mazya.pdf, discusses inequality between the $L^p$ norm in a cube (or a ball) of a function and its gradient, assuming the function vanishes on some compact subset $F$ of the cube. For instance, Theorem 9.1 gives a nice inequality between these quantities involving the $p$-capacity of the set $F$ if $1 < p < n$. My question is what happens if $p > n$ and if $p = n$. If $n > p$, it is known that every non-empty subset of the cube is an essential subset (An essential subset is a non-negligible subset. A negligible subset is a subset that verifies (9.7) for a sufficiently small constant $\gamma$). Therefore, the situation is clearly different. I just can not get how to prove a similar inequality in such case. I would expect something like the following to hold if $p>n$:

$$ \int_{Q_d} |u(x)|^p dx \leq C d^p \int_{Q_d} |\nabla u|^p dx, $$

where $C$ is a positive constant and $d$ is the edge of the cube.

Answer 1

For $p > n$, assume that the Hölder continuous representative vanishes at some point $a \in Q^d$. Sinco $p > n$, by the Sobolev--Morrey embedding, for every $x \in Q^d$ $$ \vert u (x) \vert= \vert u (x) - u (a)\vert \le C \Bigl(\int_{Q_d} \lvert D u\rvert^p\Bigr)^\frac{1}{p} \lvert x - a \rvert^{1 - \frac{d}{p}}. $$ By integration this implies that $$ \int_{Q^d} \vert u (x) \vert^p\,dx \le C \Bigl(\int_{Q_d} \lvert D u\rvert^p\Bigr)\Bigl(\int_{Q^d} \lvert x - a \rvert^p \,dx\Bigr) \le C' d^p \int_{Q_d} \lvert D u\rvert^p\Bigr. $$