Compact-manifold Dirichlet spectrum and boundary regularity

On each of finitely many relatively compact coordinate charts, metric eigenvalues and the volume density have positive lower and finite upper bounds. Thus the intrinsic $H^1$ norm is equivalent to the sum of coordinate $H^1$ norms of a fixed smooth partition of unity. A Cauchy sequence has $L^2$ limits for the localized functions and their first derivatives; testing against a smooth function and passing to the limit identifies the derivative limits as the weak derivatives of the function limit. This proves completeness; $V$, being a closure, is closed. Moreover $1\le q+c\le 1+2\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$, so $b$ and the $H^1$ inner product have equivalent norms. To prove density of $V$ in $H$, localize an $H$-function by a finite smooth partition. In an interior chart, extend each localized piece by zero and approximate it in $L^2$ by mollification. In a boundary half-chart, the localized piece is supported away from the chart sides; multiply it by a smooth cutoff that vanishes for $0\le x_n\le \delta$ and equals one for $x_n\ge 2\delta$. The boundary face has $n$-dimensional volume zero. Given $ϵ>0$, choose $R$ so large that ${\int }_{\{\mathrm{\mid }u{\mathrm{\mid }}^2>R\}}\mathrm{\mid }u{\mathrm{\mid }}^{2}d{V}_g<ϵ\mathrm{/}2$, by truncating the integrable function $\mathrm{\mid }u{\mathrm{\mid }}^2$. For the strip $S_{\delta }=\{0<x_n<2\delta \}$, ${\int }_{S_{\delta }}\mathrm{\mid }u{\mathrm{\mid }}^{2}d{V}_g\le ϵ\mathrm{/}2+R{\mathrm{vol}}_g(S_{\delta })$, and ${\mathrm{vol}}_g(S_{\delta })=O(\delta )$ because the coordinate density is bounded. Thus the $L^2$ error tends to zero as $\delta \downarrow 0$. Each cut-off piece is supported away from the boundary and is then approximated by mollification. Summing the finite chart approximations proves that $C_c^{\mathrm{\infty }}(M^{\circ })$ is dense in $H$.

The inclusion $V\hookrightarrow H$ is compact without an arbitrary-index choice argument. Indeed, a localized $v\in V$ extends by zero across a flattened boundary: approximate in $H^1$ by localized interior smooth functions, extend those, and pass to the weak-derivative identities. All extensions have bounded support in a fixed coordinate box. On a cube $C$ of side $\delta$, successively average in its $n$ coordinate variables. The one-variable identity $f(s)-f(t)={\int }_t^s{f}^{\mathrm{\prime }}(r)dr$ and Cauchy–Schwarz give $\mathrm{\parallel }f-{\mathrm{avg}}_{C}f{\mathrm{\parallel }}_{L^2(C)}\le n\delta \mathrm{\parallel }Df{\mathrm{\parallel }}_{L^2(C)}$ by telescoping those averages; density extends this to $H^1$. Replacement by the mean on each cube of a fixed finite grid is therefore a finite-rank approximation to each coordinate inclusion, with error $O(\delta )$ uniformly on the $H^1$ unit ball. Sum the finitely many coordinate reconstructions to approximate $V\hookrightarrow H$ in operator norm by finite-rank maps. A bounded finite-dimensional set has finite nets; choosing $\delta =2^{-k}$ makes the inclusion's image totally bounded, and completeness of $H$ makes its closure compact. Only countably many finite nets are involved.

In a coordinate chart $Av=f$ is $-{\mathrm{\partial }}_i(\rho g^{ij}{\mathrm{\partial }}_{j}v)+\rho qv=\rho f,\rho =\sqrt{\det g}\mathrm{.}$ The principal matrix is smooth uniformly positive on each compact chart. Localize by a smooth cutoff; its commutator with $A$ has order one. For $n=1$, put $G=\rho g^{11}>0$ on a localized interval or half-interval and absorb the first-order cutoff commutator into an $L^2$ right side $f_{\chi }$, with $\mathrm{\parallel }{f}_{\chi }{\mathrm{\parallel }}_2\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1})$. The weak equation is $-(G{v}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+\rho qv=f_{\chi }$, so $(G{v}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}=\rho qv-f_{\chi }\in L^2$. Since $G{v}^{\mathrm{\prime }}\in L^2$, this gives $G{v}^{\mathrm{\prime }}\in H^1$, and smooth positivity of $G$ gives $v\in H^2$ with the local estimate $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }{f}_{\chi }{\mathrm{\parallel }}_2+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1})$. For $n\ge 2$, in a boundary chart flatten to $x_n\ge 0$ and localize away from the chart sides. The localization can be regarded as a zero-Dirichlet function on a bounded smooth domain agreeing with the half-space near its support and rounded elsewhere. Applying [F3] and the same interior estimate, then summing the finite cover, gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1})$ in every dimension. Testing the equation with $v$ gives $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }v{\mathrm{\parallel }}_2)$ by absorbing the gradient term, so this is the stated $k=0$ estimate.

For $f\in H$, use [F1] to solve $b(Kf,v)=(f,v{)}_H$. The map $K:H\to V$ is bounded, and step 2.1 makes $K:H\to H$ compact. Symmetry gives $(Kf,h{)}_H=b(Kf,Kh)=(f,Kh{)}_H$, positivity gives $(Kf,f{)}_H=b(Kf,Kf)>0$ for $f\mathrm{\ne }0$, and density of $V$ makes $K$ injective. Its range is dense since a vector orthogonal to the range lies in the kernel by self-adjointness. Apply [F2] to obtain positive eigenvalues ${\kappa }_j\to 0$ and a complete $H$-orthonormal family; put ${\mu }_j={\kappa }_j^{-1}$. Repeated finite multiplicities are included. The vectors $e_j\mathrm{/}\sqrt{{\mu }_j}$ are $b$-orthonormal and complete in $V$: if $b(v,e_j)=0$ for every $j$, then $(v,e_j{)}_H=0$ for every $j$, hence $v=0$. Orthogonal projection and completeness therefore give the asserted $b$-Parseval identity and convergence of the spectral projections in $V$.

Here is the boundary induction for higher orders. For $n=1$, in each localized chart write $G=\rho g^{11}>0$ and $-(G{v}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+\rho qv=f_{\chi }$, where the cutoff commutator satisfies $f_{\chi }\in H^k$ and $\mathrm{\parallel }{f}_{\chi }{\mathrm{\parallel }}_{H^k}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+1}})$ whenever $f\in H^k$ and $v\in H^{k+1}$. Then $(G{v}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}=\rho qv-f_{\chi }\in H^k$, so $G{v}^{\mathrm{\prime }}\in H^{k+1}$ and smooth positivity of $G$ gives $v\in H^{k+2}$, with $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+2}}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+1}})$. This proves the one-dimensional induction, including its $k=0$ start. For $n\ge 2$, suppose $f\in H^k$ and $v\in H^{k+1}$ is already known. In a flattened boundary chart apply a tangential derivative $D_{x^{\mathrm{\prime }}}^{\alpha }$, $\mathrm{\mid }\alpha \mathrm{\mid }=k$, to the localized equation. Its commutator with the second-order operator contains derivatives of $v$ of order at most $k+1$ with bounded smooth coefficients. The localized $D_{x^{\mathrm{\prime }}}^{\alpha }v$ has zero trace and lies in $H^1$; this follows first for tangential difference quotients, which preserve zero trace, and then by their $H^1$ limit. The $k=0$ local estimate of step 2.2 gives its $H^2$ bound in terms of $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+1}}$. If a derivative has total order $k+2$ and at least three normal factors, differentiate the equation $k$ times, isolating the term $\rho g^{nn}{D}_n^{2}v$. Divide by the positive coefficient $\rho g^{nn}$. Each other highest-order derivative has fewer normal factors because its two principal derivatives include a tangential derivative; all commutators have order at most $k+1$. Induction on the number of normal factors therefore bounds every derivative of order $k+2$. Interior patches allow derivatives in all directions. Difference quotients justify each differentiation before regularity is established; the bounds are uniform in their increments, and testing their weak limits identifies the claimed derivatives. Nested cutoffs and the finite cover give $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+2}}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }v{\mathrm{\parallel }}_{H^{k+1}})$, and the previous induction level absorbs the last term into $C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }v{\mathrm{\parallel }}_2)$. This proves all asserted elliptic estimates.

For completeness the needed high-order embedding also has a finite-chart proof with this axiom base. In a half-chart extend across $x_n=0$ by ${\sum }_{j=1}^{k+1}{a}_{j}v(x^{\mathrm{\prime }},-j{x}_n)$ for $x_n<0$, where the finite Vandermonde system ${\sum }_j{a}_j(-j{)}^{\mathrm{\ell }}=1$ for $0\le \mathrm{\ell }\le k$ matches the derivatives at the interface. After a cutoff this is a bounded $H^k$ extension; the assertion follows first on smooth functions and then by the weak derivative identities and approximation. On a bounded support box, iterated one-dimensional integration gives $\mathrm{\parallel }{D}^{\alpha }w{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\mathrm{\parallel }{D}_1\cdots D_n{D}^{\alpha }w{\mathrm{\parallel }}_2$ for smooth compactly supported $w$. Apply this also to first derivatives of $D^{\alpha }w$ and to differences of mollifications. For $k\ge n+a+1$ these mollifications and all derivatives through order $a$ are uniformly Cauchy and their limits are continuous, with the asserted norm bound; passing to the coordinate segment integral identifies them as classical derivatives. Sum the finite chart extensions. This proves the embedding and completes the lemma.