Lemma 7.8.21BE242B641AA

Proof. We reduce to the Laplacian, treat the case $F=0$ by a maximum principle argument, and reduce the general case to it.

Step 1. Reduction to $L=\Delta $. Let $C=(c_{kl})$, let $P$ be the positive definite square root of $C^{-1}$, so that $PCP=I$, and put $\tilde U=P(U)$, $\tilde w=w\circ P^{-1}$ and $\tilde F=F\circ P^{-1}$. If $q$ is a $C^2$ function near $x\in U$ and $\tilde q=q\circ P^{-1}$, then $\partial _ k\partial _ lq(x)=\sum _{a,b}P_{ak}P_{bl}(\partial _ a\partial _ b\tilde q)(Px)$ by the chain rule, and hence

The map $q\mapsto \tilde q$ is a bijection from the upper test functions for $w$ at $x$ onto those for $\tilde w$ at $Px$, so $\tilde w$, $\Delta $ and $\tilde F$ satisfy the hypothesis of the lemma on $\tilde U$. Suppose that the lemma holds for them. Then $w=\tilde w\circ P$ is locally integrable, and for every nonnegative test function $\chi $ on $U$ the change of variables $y=Px$, with $\tilde\chi =\chi \circ P^{-1}$ and $L\chi =(\Delta \tilde\chi )\circ P$, gives

the first equality is the definition of distributional derivatives. We may therefore assume that $L=\Delta $.

Step 2. The case $F=0$. We show that $w$ satisfies assertion (1) of theorem 3.4.5 for every ball $D=B(y,r)$ with $\overline{B(y,r)}\subset U$; by the converse part of that theorem, $w$ is then subharmonic on $U$. Let $h$ be a real function which is harmonic on $B(y,r)$ and continuous on $\overline{B(y,r)}$, with $\varlimsup _{B(y,r)\ni x\to \zeta }w(x)\le h(\zeta )$ for every $\zeta \in \partial B(y,r)$, and suppose that $w(x_1)>h(x_1)$ for some $x_1\in B(y,r)$. Choose $\varepsilon >0$ so small that the upper semicontinuous function $g(x)=w(x)-h(x)-\varepsilon (r^2-|x-y|^2)$ on $B(y,r)$ satisfies $g(x_1)>0$.

Since $h$ is continuous on the closed ball, $\varlimsup _{B(y,r)\ni x\to \zeta }g(x)\le 0$ for every $\zeta \in \partial B(y,r)$. Hence every $\zeta \in \partial B(y,r)$ has an open neighborhood $N_\zeta $ with $g<g(x_1)/2$ on $N_\zeta \cap B(y,r)$; finitely many of these neighborhoods cover the sphere, and the complement $K$ of their union in $\overline{B(y,r)}$ is a compact subset of $B(y,r)$ containing $x_1$. The upper semicontinuous function $g$ attains its maximum over $K$ at a point $x_0$, and $c=g(x_0)\ge g(x_1)>0$; since $g<g(x_1)/2$ outside $K$, we have $g\le c$ on $B(y,r)$. The function $q=h+\varepsilon (r^2-|x-y|^2)+c$ is smooth on $B(y,r)$, satisfies $q-w=c-g\ge 0$ there and $q(x_0)=w(x_0)$, so it is an upper test function for $w$ at $x_0$. But $\Delta q=-2d\varepsilon <0$, which contradicts the hypothesis. Hence $w\le h$ on $B(y,r)$, and $w$ is subharmonic on $U$. Being real-valued, it is not identically $-\infty $ on any connected component of $U$, so theorem 3.4.12(1) shows that $w$ is locally integrable and $\Delta w\ge 0$.

Step 3. The general case. Let $V\Subset U$ be open. Every compact subset of $U$ lies in such a set $V$, so it suffices to show that $w$ is locally integrable on $V$ and that $\Delta w\ge F$ on $V$. Choose an open set $V'$ with $\overline V\subset V'\Subset U$. By lemma 2.5.1(1), the mollifications of $F$ converge to $F$ uniformly on $\overline{V'}$; subtracting from each of them its maximal deviation from $F$ on $\overline{V'}$, we obtain smooth functions $F_ j$ on a neighborhood of $\overline{V'}$ such that $F_ j\le F$ on $\overline{V'}$ and $F_ j\to F$ uniformly there. Let $\vartheta \in C^\infty _ c(V')$ be equal to one on an open set $W\supset \overline V$ (lemma 2.5.1(4)). By proposition 3.1.13, the convolution $h_ j=E_ d*(\vartheta F_ j)$, where $\vartheta F_ j$ is extended by zero, is a smooth function on $\mathbb R^ d$ with $\Delta h_ j=\vartheta F_ j$; in particular $\Delta h_ j=F_ j$ on $W$.

If $q$ is an upper test function for $w-h_ j$ at $x\in V$, then $q+h_ j$ is an upper test function for $w$ at $x$, so that $\Delta q(x)+F_ j(x)\ge F(x)$, that is, $\Delta q(x)\ge F(x)-F_ j(x)\ge 0$. Step 2, applied to the real upper semicontinuous function $w-h_ j$ on $V$, shows that $w-h_ j$ is locally integrable on $V$ and $\Delta (w-h_ j)\ge 0$ there. Hence $w$ is locally integrable on $V$ and $\Delta w\ge F_ j$ on $V$. Testing against a nonnegative test function on $V$ and letting $j\to \infty $, we obtain $\Delta w\ge F$ on $V$. $\square$