Lemma 3.4.1744B1F1B5F95E

Proof. We first record a local estimate. Let $a\in \Omega $ and $r>0$ with $\overline{B(a,4r)}\subset \Omega $, and let $M\geq 0$ be a common upper bound of the functions $u_ j$ on $B(a,4r)$. Suppose that $u_ j(z)\geq -C$ for some index $j$, some point $z\in B(a,r)$ and some constant $C$. Then $B(a,r)\subset B(z,2r)$ and $\overline{B(z,2r)}\subset B(a,4r)$. Since $M-u_ j\geq 0$ on $B(z,2r)$, the ball submean inequality of theorem 3.4.5 gives

and since $|u_ j|\leq (M-u_ j)+M$, we obtain

Passing to a subsequence, we may assume that $x_ j\to x_0\in K$. Choose $r>0$ with $\overline{B(x_0,4r)}\subset \Omega $. Discarding finitely many terms, we may assume that $|x_ j-x_0|<r$ for all $j$. Then (3.4.18), with $a=x_0$, $z=x_ j$ and $C=C_0$, bounds $\int _{B(x_0,r)}|u_ j|\, \mathrm d x$ uniformly in $j$.

Let $Y$ be the set of points $y\in \Omega $ that have a neighborhood $U$ with $\sup _ j\int _ U|u_ j|\, \mathrm d x<\infty $. Then $Y$ is open and contains $x_0$. Let $y\in \Omega $ lie in the closure of $Y$. Choose $r>0$ with $\overline{B(y,4r)}\subset \Omega $ and an open ball $A$ with $\overline A\subset Y\cap B(y,r)$. Covering the compact set $\overline A$ by finitely many of the neighborhoods in the definition of $Y$, we find $C_ A=\sup _ j\int _ A|u_ j|\, \mathrm d x<\infty $. Put $C=C_ A/|A|$. For each $j$ there is a point $z_ j\in A$ with $u_ j(z_ j)\geq -C$, since otherwise $\int _ A|u_ j|\, \mathrm d x\geq \int _ A(-u_ j)\, \mathrm d x>C|A|=C_ A$. By (3.4.18), with $a=y$ and $z=z_ j$, the integrals $\int _{B(y,r)}|u_ j|\, \mathrm d x$ are bounded uniformly in $j$, so $y\in Y$. Thus $Y$ is closed in $\Omega $, and $Y=\Omega $ because $\Omega $ is connected. Finally, every compact subset of $\Omega $ is covered by finitely many of the neighborhoods in the definition of $Y$. $\square$