Theorem 3.4.19F159A5A96A98

Proof. We prove (2) first, because (1) reduces to it. As observed after theorem 3.4.12, $T$ is real, $\Delta T\geq 0$, and $T$ has a canonical representative $u$. We begin by showing that $(u_ j)$ is bounded in $L^1(L)$ for every compact set $L\subset \Omega $. Fix such a set $L$, let $\chi $ be a nonnegative test function equal to one near $L$, and let $M\geq 0$ bound all $u_ j$ from above on $\operatorname{Supp}\chi $. Then $|u_ j|\leq 2M-u_ j$ on $\operatorname{Supp}\chi $, so

which is bounded because $\int u_ j\chi \, \mathrm d x$ converges.

Next we show that the mollifications $u_ j*\rho _\varepsilon $ converge locally uniformly. Fix $\varepsilon >0$. For $x\in \Omega _\varepsilon $, the function $\rho _\varepsilon (x-\cdot )$ is a test function, so $u_ j*\rho _\varepsilon (x)\to \langle T,\rho _\varepsilon (x-\cdot )\rangle =u*\rho _\varepsilon (x)$. Let $L\subset \Omega _\varepsilon $ be compact; then $L'=\{ y:\operatorname {dist}(y,L)\leq \varepsilon \} $ is a compact subset of $\Omega $, and the mean value theorem, applied to $\rho _\varepsilon $, gives

Thus the functions $u_ j*\rho _\varepsilon $ are $A$-Lipschitz on $L$, where $A=\sup |\nabla \rho _\varepsilon |\cdot \sup _ j\int _{L'}|u_ j|\, \mathrm dy$, and so is their pointwise limit $u*\rho _\varepsilon $. Given $\tau >0$, cover $L$ by finitely many balls $B(x_ i,\tau /(3A+1))$ with $x_ i\in L$, and choose $j_0$ such that $|u_ j*\rho _\varepsilon (x_ i)-u*\rho _\varepsilon (x_ i)|<\tau /3$ for all $i$ and all $j\geq j_0$. Then $|u_ j*\rho _\varepsilon -u*\rho _\varepsilon |<\tau $ on $L$ for $j\geq j_0$, so the convergence is uniform on $L$.

Now let $\chi $ be a nonnegative test function, choose $\varepsilon $ so small that $\operatorname{Supp}\chi \subset \Omega _\varepsilon $, let $\delta >0$, and put $g=u*\rho _\varepsilon +\delta $. By (3.4.13), $u_ j\leq u_ j*\rho _\varepsilon $ and $u\leq u*\rho _\varepsilon $ on $\Omega _\varepsilon $. By the uniform convergence just proved, $u_ j\leq g$ on $\operatorname{Supp}\chi $ for all large $j$; also $u\leq g$. Hence $|u_ j-u|\leq (g-u_ j)+(g-u)$ almost everywhere on $\operatorname{Supp}\chi $, and since $\int u_ j\chi \, \mathrm d x\to \int u\chi \, \mathrm d x$,

Let $\delta \downarrow 0$, and then $\varepsilon \downarrow 0$; since $u*\rho _\varepsilon \to u$ in $L^1$ on $\operatorname{Supp}\chi $ (lemma 2.5.1), the right-hand side tends to zero. Choosing $\chi $ equal to one on a prescribed compact set proves $L^1$ convergence on that set. This proves (2).

(1) Suppose that $u_ j$ does not tend to $-\infty $ uniformly on compact subsets. Then there are a compact set $K\subset \Omega $, a constant $C_0$ and infinitely many indices $j$ with $\sup _ Ku_ j>-C_0$, and for these $j$ we can choose $x_ j\in K$ with $u_ j(x_ j)>-C_0$. By lemma 3.4.17, a subsequence, which we again denote by $(u_ j)$, is bounded in $L^1(L)$ for every compact set $L\subset \Omega $; in particular, every $u_ j$ is locally integrable. By (2), it suffices to show that a further subsequence converges in the sense of distributions.

Choose open sets $U_ m\Subset \Omega $ increasing to $\Omega $, for instance the interiors of the sets of a compact exhaustion (section 2.2), and constants $M_ m\geq 0$ with $u_ j\leq M_ m$ on $\overline{U_ m}$ for all $j$. On $U_ m$, the Borel measures $(M_ m-u_ j)\, \mathrm d x$ are positive, and their masses on a compact set $L\subset U_ m$ are at most $M_ m|L|+\sup _ j\int _ L|u_ j|\, \mathrm dx$, uniformly in $j$; in particular, they are Radon measures by proposition 2.2.2. By theorem 2.6.5 and a diagonal extraction over $m$, we may assume that for each $m$ these measures converge vaguely on $U_ m$. Every $\chi \in C_ c(\Omega )$ has its support in some $U_ m$, so

converges. Let $T(\chi )$ denote the limit when $\chi $ is a test function. If $\operatorname{Supp}\chi \subset L$ with $L$ compact, then $|T(\chi )|\leq \sup _ j\int _ L|u_ j|\, \mathrm d x\cdot \sup |\chi |$. Hence $T$ is a distribution of order zero, and $u_ j\to T$ in the sense of distributions. $\square$