Weak compact-domain comparison for continuous nonnegative potentials

At an interior chart point $p$, replace $w$ by its quadratic Taylor polynomial $P$ in that chart. The metric dual of $dw$ has components $h^{ij}{\mathrm{\partial }}_{j}w$; its first derivatives depend only on the derivatives of $w$ through order two. Consequently its covariant derivative and trace at $p$ agree with those of ${\mathrm{grad}}_{h}P$. Applying the smooth formulas of [F2] to $P$ proves ${\mathrm{\Delta }}_{h}w=m^{-1}{\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_{j}w)$ at $p$, with $A^{ij}=m{h}^{ij}$, since the right side also depends only on that same two-jet. Thus the classical hypothesis gives $-{\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_{j}w)+mVw\le 0$ throughout the chart. Multiply by any nonnegative $\phi \in C_c^{\mathrm{\infty }}$ of the chart and integrate. Integration of the derivative of the compactly supported $C^1$ product $A^{ij}{\mathrm{\partial }}_{j}w\phi$, first in the $i$th coordinate, gives $\int (A^{ij}{\mathrm{\partial }}_{j}w{\mathrm{\partial }}_i\phi +mVw\phi )dx\le 0$ by the fundamental theorem of calculus. No derivative of $V$ is taken.

Fix $ϵ>0$ and put $z=(w-ϵ{)}^{+}$, $K=\{w\ge ϵ\}$ and $K^{\mathrm{\prime }}=\{w\ge ϵ\mathrm{/}2\}$, taking these closed sets in $\bar{D}$. They are compact subsets of $D$ because the boundary values are nonpositive; in particular $\mathrm{supp}z\subseteq K$. If $K$ is empty there is nothing to prove for this $ϵ$. Otherwise cover $K^{\mathrm{\prime }}$ by finitely many relatively compact interior coordinate balls $U_1,\dots ,U_N$, whose closures lie in larger interior charts, and set $U_0=D\setminus K^{\mathrm{\prime }}$. By [F1] and [F4], this finite open cover of the smooth manifold $D$ has a subordinate partition ${\varphi }_0,\dots ,{\varphi }_N$. Define ${\eta }_j={\varphi }_j\mathrm{/}({\sum }_{\mathrm{\ell }=0}^N{\varphi }_{\mathrm{\ell }}^2{)}^{1\mathrm{/}2}$. The denominator is positive everywhere. Each ${\eta }_j$ is smooth, ${\sum }_{j=0}^N{\eta }_j^2=1$, and, for $j\ge 1$, $\mathrm{supp}{\eta }_j$ is compactly contained in $U_j$. On the open neighborhood $\{w>ϵ\mathrm{/}2\}$ of $K$, ${\eta }_0=0$, so ${\sum }_{j=1}^N{\eta }_j^2=1$ and ${\sum }_{j=1}^{N}2{\eta }_{j}d{\eta }_j=0$ there.

On each $U_j$, $j\ge 1$, the coordinate representative of $w$ is in $H^1$: $w$ and its first derivatives are bounded on its compact closure in the larger chart, and the coordinate ball has finite measure. The source $-mVw$ is locally bounded and hence locally square integrable. Thus [F3] extends the weak inequality of step 1.1 to the nonnegative test ${\eta }_j^{2}z$, with $Dz={\mathbf{1}}_{\{w>ϵ\}}Dw$ and $D({\eta }_j^{2}z)={\eta }_j^{2}Dz+2{\eta }_{j}zD{\eta }_j$. The coefficients are bounded on the test support. Rewrite these chart integrals using the density in [F2] and sum over $j=1,\dots ,N$. Since $z$ and $Dz$ vanish off $K$ up to null sets, the two sum identities of step 1.2 cancel the cross terms and give ${\int }_D\mathrm{\mid }\mathrm{\nabla }z{\mathrm{\mid }}_h^{2}d{\mu }_h+{\int }_{D}Vwzd{\mu }_h\le 0$.

Both integrands in step 2.1 are nonnegative, since $V\ge 0$ and $w>ϵ$ wherever $z>0$. Hence $\mathrm{\nabla }z=0$ almost everywhere. On the open set $\{w>ϵ\}$, [F3] identifies this gradient with that of the $C^2$ function $w$. Continuity of $\mathrm{\nabla }w$ and positivity of the volume density therefore give $\mathrm{\nabla }w=0$ everywhere on that open set. On every coordinate ball contained in it, the fundamental theorem of calculus along segments shows that $w$ is constant. Thus $w=c>ϵ$ on each connected component $G$ of $\{w>ϵ\}$. If $p$ is in the closure of $G$ relative to $D$, continuity gives $w(p)=c>ϵ$; a small connected coordinate ball about $p$ in $\{w>ϵ\}$ meets $G$, so $p\in G$. Thus $G$ is both open and closed in $D$ and is a whole connected component of $D$. Its closure meets $\mathrm{\partial }D$, where continuity would give $w=c>ϵ$, contrary to the boundary condition. Hence no such $G$ exists, and $w\le ϵ$. Letting $ϵ$ decrease to zero proves $w\le 0$. The empty-domain case is vacuous; a nonempty zero-dimensional domain cannot satisfy the component-boundary hypothesis. Finally [F5] gives the stated $C^{2,\alpha }(\bar{D})$ consequence. Only local boundedness and nonnegativity of the continuous potential were used in the argument.