Poincaré-type Inequality

Question

In Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" one finds the following remark:

Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\nabla u\|_{p} \leq 1 .$ Set $k=1+\|u\|_{p}^{-p}\left(\text { for }\|u\|_{p} \leq \infty\right)$. Let $B_{x}$ denote the unitary ball in $\mathbb{R}^N$ centered at $x$, and let $\beta_x$ be its characteristic function. Clearly there is some $x$ such that \begin{equation} \label{lb} \int|\nabla u|^{p} \beta_{x}<k \int|u|^{p} \beta_{x}. \end{equation}

I can't see why this inequality holds.

Answer 1

For $k$ to make sense, we should assume that $\|u\|_p\ne0$. Let $l(x)$ and $r(x)$ denote the left- and right-hand sides of your displayed inequality. Then, by Tonelli's theorem, for any real $p>0$ $$\int dx\,l(x)=\int dx\int dz\, 1\{|z-x|<1\}|\nabla u(z)|^p \\ =\int dz\,|\nabla u(z)|^p\int dz\, 1\{|z-x|<1\} =v_n\|\nabla u\|_p^p\le v_n,$$ where $v_n$ is the volume of the unit ball in $\mathbb R^n$. Similarly, $$\int dx\,r(x) =kv_n\|u\|_p^p>v_n.$$ So, $$\int dx\,l(x)<\int dx\,r(x)$$ and hence $$l(x)<r(x)$$ for at least one $x$, as desired.