Compact-domain comparison with a nonnegative potential

If $D$ is empty the assertion is vacuous. In dimension zero the manifold is discrete, so each component of $D$ is a closed singleton and cannot have closure meeting $\mathrm{\partial }D$; thus the hypotheses force $D$ to be empty. In positive dimension, $N\cap \mathrm{\partial }M\subseteq \mathrm{\partial }N$. Indeed, if an ambient-boundary point were an intrinsic interior point of $N$, an ambient boundary coordinate $r\ge 0$ would restrict to a smooth function on an open neighborhood in $N$ with an interior minimum zero. Its differential there would vanish. But the inclusion of the full-dimensional embedded $N$ has an invertible tangent map, so the nonzero differential $dr$ of that boundary coordinate would restrict to a nonzero differential, a contradiction. Hence $D\setminus \mathrm{\Omega }\subseteq \mathrm{\partial }D$, and $N\setminus \mathrm{\partial }D\subseteq \mathrm{\Omega }$ by the smooth-domain convention.

Fix an interior point $p\in \mathrm{\Omega }$ and replace $w$ in its coordinates by the polynomial with the same value and first and second derivatives at $p$. Metric duality gives $({\mathrm{grad}}_{h}w{)}^i=h^{ij}{\mathrm{\partial }}_{j}w$. Its first derivatives depend only on the two-jet of $w$, so the trace of its covariant derivative agrees at $p$ with that of the smooth polynomial. The density-divergence formula for a $C^1$ vector field follows likewise by replacing its coordinate coefficients by affine functions with the same one-jet. Consequently the smooth formulas in [F1] imply ${\mathrm{\Delta }}_{h}w=m^{-1}{\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_{j}w)$ for the stated $C^2$ function in every interior chart.

In a coordinate ball $U=x^{-1}(B)$ as in [F2], the differential inequality becomes $-{\mathrm{\partial }}_i(A^{ij}{\mathrm{\partial }}_{j}w)+qw\le 0$. Multiply by a nonnegative $\phi \in C_c^{\mathrm{\infty }}$ and integrate in chart coordinates. Each product $A^{ij}{\mathrm{\partial }}_{j}w\phi$ is $C^1$ and compactly supported, so extension by zero and one-dimensional integration of its $i$th derivative give $\int (A^{ij}{\mathrm{\partial }}_{j}w{\mathrm{\partial }}_i\phi +qw\phi )dx\le 0$. This has local source $-qw\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$. Thus [F3] extends it to each nonnegative test $\phi ={\eta }^2{v}_{ϵ}$, with the stated truncation and product derivative formulas. All tests are supported in the interior, so no manifold-boundary integration or boundary-chart approximation occurs.

Fix $ϵ>0$. The set $K=\{x\in \bar{D}:w(x)\ge ϵ\}$ is compact and disjoint from $\mathrm{\partial }D$, so step 1.1 gives $K\subset \mathrm{\Omega }$ and $\mathrm{supp}{v}_{ϵ}\subseteq K$ (support taken in $N$). If $K$ is empty then $v_{ϵ}=0$. Otherwise take a locally finite partition $({\chi }_a)$ from [F2], subordinate to the coordinate balls $U=x^{-1}(B)$ of [F2], and put $s=({\sum }_a{\chi }_a^2{)}^{1\mathrm{/}2}$ and ${\eta }_a={\chi }_a\mathrm{/}s$. The locally finite sum is smooth and strictly positive because ${\sum }_a{\chi }_a=1$; each ${\eta }_a$ is smooth, has compact support in its chart, and ${\sum }_a{\eta }_a^2=1$. Only finitely many supports meet $K$. Apply step 2.1 with each corresponding ${\eta }_a^2{v}_{ϵ}$ and sum. With the density from [F1] the result is ${\int }_{\mathrm{\Omega }}(⟨{\mathrm{\nabla }}_{h}w,{\mathrm{\nabla }}_h{v}_{ϵ}{⟩}_h+Vw{v}_{ϵ})d{\mu }_h\le 0$: the terms containing partition derivatives cancel because ${\sum }_a{\mathrm{\nabla }}_h({\eta }_a^2)=0$ near $K$, and $v_{ϵ}$ and its weak derivatives vanish off $K$.

The integrand in step 3.1 equals ${\mathbf{1}}_{\{w>ϵ\}}\mathrm{\mid }{\mathrm{\nabla }}_{h}w{\mathrm{\mid }}_h^2+Vw{v}_{ϵ}$ almost everywhere. Both terms are nonnegative, so the weak gradient of $v_{ϵ}$ is zero almost everywhere. On each coordinate ball $U=x^{-1}(B)$ of [F2], smooth positive definiteness and the positive density imply equivalence with zero Euclidean weak derivatives. Apply [F4] on that ball. Its almost everywhere constant value is its value everywhere by continuity of $v_{ϵ}$, since every nonempty coordinate ball has positive measure. Thus $v_{ϵ}$ is locally constant on $\mathrm{\Omega }$. At any $q\in D\setminus \mathrm{\Omega }$, step 1.1 places $q$ in $\mathrm{\partial }D$, so $w(q)\le 0<ϵ$. Continuity supplies a relative neighborhood of $q$ in $D$ on which $w<ϵ$, and $v_{ϵ}=0$ there. Consequently $v_{ϵ}$ is locally constant on all of $D$, and connectedness makes it a single constant on each component $C$. Choose $p\in \bar{C}\cap \mathrm{\partial }D$; continuity on $\bar{D}$ and $w(p)\le 0$ show that this constant is $v_{ϵ}(p)=0$. Hence $v_{ϵ}=0$ throughout $D$ for every $ϵ>0$, and $w\le 0$. For a subsolution $u_{-}$ and supersolution $u_{+}$ satisfying $({\mathrm{\Delta }}_h-V)u_{-}\ge 0$, $({\mathrm{\Delta }}_h-V)u_{+}\le 0$ and ordered boundary values, apply the established assertion to $w=u_{-}-u_{+}$, whose operator and boundary signs are exactly those required.