Boundary quasimode jets and exact-solution correction

Choose boundary normal coordinates $(y,x_n)$ near $x_0$, with $x_n=\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })$ and inward unit normal ${\mathrm{\partial }}_{x_n}$. The induced metric on the boundary is smooth. Choose a smooth hypersurface $S$ in $\mathrm{\partial }\mathrm{\Omega }$ through $x_0$, orthogonal there to ${\xi }_0$ (if $n=2$, take $S=\{x_0\}$). The boundary geodesics starting normally from $S$ give, by smooth dependence of the geodesic ODE and the inverse function theorem, coordinates $(s,z)$ near $x_0$; define $\varphi (s,z)=s$. Gauss' lemma gives $\mathrm{\mid }d\varphi {\mathrm{\mid }}_{g_{\mathrm{\partial }\mathrm{\Omega }}}=1$ and $d\varphi (x_0)={\xi }_0$. Shrink the patch so these coordinates and $\chi$ are supported inside it.

In boundary normal coordinates, $g^{nn}=1$ and $g^{n\alpha }=0$ at the boundary. Set $\mathrm{\Phi }{\mathrm{\mid }}_{x_n=0}=\varphi$ and ${\mathrm{\partial }}_n\mathrm{\Phi }{\mathrm{\mid }}_{x_n=0}=i$. Then $g^{ij}{\mathrm{\partial }}_i\mathrm{\Phi }{\mathrm{\partial }}_j\mathrm{\Phi }=0$ at $x_n=0$ because $\mathrm{\mid }d\varphi \mathrm{\mid }=1$. Differentiate this eikonal equation successively in $x_n$. At each order the newest unknown normal jet ${\mathrm{\partial }}_n^{j+1}\mathrm{\Phi }{\mathrm{\mid }}_{x_n=0}$ occurs linearly with coefficient $2i{g}^{nn}=2i\mathrm{\ne }0$; all other terms use already assigned jets and derivatives of $g$. Solve recursively for every normal jet. Realize the resulting jet sequence by the following local Borel construction: if $c_j(y)$ is the prescribed $j$th normal derivative, choose a fixed smooth cutoff $\theta$ equal to one near zero and supported in $(-1,1)$, and set $\mathrm{\Phi }(y,x_n)={\sum }_{j\ge 0}\frac{c_j(y)}{j!}{x}_n^j\theta (x_n\mathrm{/}{\epsilon }_j)\mathrm{.}$ For all sufficiently large $j$, choose ${\epsilon }_j<2^{-j}$ so that every derivative of order at most $j\mathrm{/}2$ of the $j$th summand has sup norm at most $2^{-j}$; this is possible because those derivatives are bounded by constants times ${\epsilon }_j^{j-m}$ with $m\le j\mathrm{/}2$. Choose any positive ${\epsilon }_j$ for the finitely many smaller $j$. For each fixed derivative order, the tail then converges uniformly, so $\mathrm{\Phi }$ is smooth and has the prescribed jets. In particular, the eikonal expression vanishes to infinite order at the boundary. Since ${\mathrm{\partial }}_n\mathrm{Im}\mathrm{\Phi }=1$ there, shrinking the collar gives $\mathrm{Im}\mathrm{\Phi }\ge x_n\mathrm{/}2$.

For $P_{\gamma }=\mathrm{div}(\gamma \mathrm{\nabla })$, direct differentiation gives $P_{\gamma }(e^{i\lambda \mathrm{\Phi }}b)=e^{i\lambda \mathrm{\Phi }}[-{\lambda }^2{p}_{\gamma }b+i\lambda L_{\gamma }b+P_{\gamma }b],$ where $p_{\gamma }=\gamma g^{ij}{\mathrm{\partial }}_i\mathrm{\Phi }{\mathrm{\partial }}_j\mathrm{\Phi }$ and $L_{\gamma }b=2\gamma g^{ij}({\mathrm{\partial }}_i\mathrm{\Phi })({\mathrm{\partial }}_{j}b)+\mathrm{div}(\gamma \mathrm{\nabla }\mathrm{\Phi })b$. The phase is common to both conductivities and $p_{\gamma }=\gamma p_1$ vanishes to infinite order on the boundary by step 2.1. Prescribe $b_{0,\gamma }{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=\chi$ and $b_{-j,\gamma }{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$ for $j\ge 1$. In boundary normal coordinates the coefficient of the new normal derivative in $L_{\gamma }b$ is $2i\gamma$, which is nonzero. Therefore the normal derivatives of $b_{0,\gamma }$ can be chosen successively so that $L_{\gamma }{b}_{0,\gamma }=0$ to infinite order at the boundary; after choosing $b_{0,\gamma },\dots ,b_{-(j-1),\gamma }$, the same recursion chooses all jets of $b_{-j,\gamma }$ so that $i{L}_{\gamma }{b}_{-j,\gamma }+P_{\gamma }{b}_{-(j-1),\gamma }=0$ to infinite order. Realize each sequence by the Borel series in step 2.1 and multiply by a fixed collar cutoff equal to one near the boundary support. Repeat for $\tilde{\gamma }$.

Let $b_{\lambda ,\gamma }={\sum }_{j=0}^K{\lambda }^{-j}{b}_{-j,\gamma }$, with the same construction for $\tilde{\gamma }$. The order-${\lambda }^2$ eikonal term and the order-$\lambda ,\dots ,{\lambda }^{1-K}$ transport terms in step 3.1 vanish to infinite order at $x_n=0$. Thus each eikonal or canceled transport coefficient is $O(x_n^R)$ for every prescribed integer $R$; the final term ${\lambda }^{-K}{P}_{\gamma }{b}_{-K,\gamma }$ has a fixed smooth bounded coefficient and is treated by increasing $K$. On the collar, $\mathrm{\mid }{e}^{i\lambda \mathrm{\Phi }}\mathrm{\mid }\le e^{-\lambda x_n\mathrm{/}2}$. For every fixed $a,R$, direct integration gives $\mathrm{\parallel }{\lambda }^a{x}_n^R{e}^{-\lambda x_n\mathrm{/}2}{\mathrm{\parallel }}_{L^2(0,\delta )}=O({\lambda }^{a-R-1\mathrm{/}2})\mathrm{.}$ Choose $R$ and then $K$ large enough to make every residual term $O({\lambda }^{-N})$. Derivatives of the collar cutoff are supported where $x_n\ge c>0$ and hence contribute $O(e^{-c\lambda })$. The same Gaussian integration shows $\mathrm{\parallel }{v}_{\lambda ,\gamma }{\mathrm{\parallel }}_{L^2}=O({\lambda }^{-1})$ and bounds its gradient norm above. Thus the $H^1$ norms are bounded above and the residual bounds hold for the arbitrary prescribed $N$.

The compact smooth manifold $\bar{\mathrm{\Omega }}$ has finitely many connected components: connected coordinate neighborhoods form an open cover and a finite subcover meets every component. Each component has nonempty boundary, since a boundaryless component of the same dimension embedded in ${\mathbb{R}}^n$ would be both open and compact. Apply [F3] on each component and patch the finitely many solutions to obtain the exact solution on $\mathrm{\Omega }$. Taking the maximum of the component Poincare constants gives $\mathrm{\parallel }z{\mathrm{\parallel }}_2\le C_P\mathrm{\parallel }\mathrm{\nabla }z{\mathrm{\parallel }}_2$ for $z\in H_0^1(\mathrm{\Omega })$. Let $r_{\lambda }=u_{\lambda ,\gamma }-v_{\lambda ,\gamma }\in H_0^1(\mathrm{\Omega })$. Its weak equation is $P_{\gamma }{r}_{\lambda }=-P_{\gamma }{v}_{\lambda ,\gamma }$. Test with $r_{\lambda }$, use $\gamma \ge {\gamma }_{\ast }>0$, and use the Poincare inequality in [F3]: ${\gamma }_{\ast }\mathrm{\parallel }\mathrm{\nabla }{r}_{\lambda }{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }{P}_{\gamma }{v}_{\lambda ,\gamma }{\mathrm{\parallel }}_2\mathrm{\parallel }{r}_{\lambda }{\mathrm{\parallel }}_2\le C_P\mathrm{\parallel }{P}_{\gamma }{v}_{\lambda ,\gamma }{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }{r}_{\lambda }{\mathrm{\parallel }}_2\mathrm{.}$ If the last gradient norm is zero then $r_{\lambda }=0$ by [F3]; otherwise divide by it. This proves $\mathrm{\parallel }{r}_{\lambda }{\mathrm{\parallel }}_{H^1}=O({\lambda }^{-N})$. The same argument applies to $\tilde{\gamma }$.

In the collar, write $J(y,x_n)dS(y)d{x}_n$ for Euclidean volume in boundary normal coordinates, with $J(y,0)=1$. Since $\mathrm{\Phi }=\varphi +i{x}_n+O(x_n^2)$, $\mathrm{Im}\mathrm{\Phi }=x_n+O(x_n^2)$ and $g(d\mathrm{\Phi },\overline{d\mathrm{\Phi }})=2+O(x_n)$. The boundary values of both leading amplitudes are $\chi$, and lower amplitudes vanish there. Differentiating $v_{\lambda ,\gamma }$ and $v_{\lambda ,\tilde{\gamma }}$ therefore gives $\mathrm{\nabla }{v}_{\lambda ,\gamma }\cdot \overline{\mathrm{\nabla }{v}_{\lambda ,\stackrel{~}{\gamma }}}=2\lambda e^{-2\lambda x_n}\mathrm{\mid }\chi (y){\mathrm{\mid }}^2+e^{-2\lambda x_n}O(1)+e^{-2\lambda x_n}O(\lambda x_n)+e^{-2\lambda x_n}O({\lambda }^2{x}_n^2),$ uniformly for $0\le x_n\le {\lambda }^{-1\mathrm{/}2}$. On this shrinking collar $\lambda x_n^2\le 1$, so after $t=\lambda x_n$ the phase remainder factor is uniformly bounded; boundedness of $F,J,\chi$ gives an integrable majorant $C{t}^k{e}^{-t}$ for the rescaled leading term. For each fixed $t$, the rescaled leading term tends to $2t^k{e}^{-2t}F(y,0)\mathrm{\mid }\chi (y){\mathrm{\mid }}^2$. Each displayed remainder has an additional factor ${\lambda }^{-1}$ after scaling and is dominated by $C{\lambda }^{-1}(t^k+t^{k+1}+t^{k+2})e^{-t}$, including the phase error $e^{O(t^2\mathrm{/}\lambda )}-1=O(t^2\mathrm{/}\lambda )$; its integral tends to zero. On the rest of the fixed collar, $\mathrm{Im}\mathrm{\Phi }\ge x_n\mathrm{/}2$ gives exponential decay $e^{-\lambda x_n}\le e^{-\sqrt{\lambda }}$; every polynomial prefactor is therefore negligible. The cutoff region away from the collar is also exponentially small. Thus the limit for approximate beams is $2{\int }_0^{\mathrm{\infty }}{t}^k{e}^{-2t}dt{\int }_{\mathrm{\partial }\mathrm{\Omega }}F\mathrm{\mid }\chi {\mathrm{\mid }}^{2}dS,$ whose constant is strictly positive.

Taking equal conductivities, $F=1$, and $k=0$ in step 5.2 gives ${\lim }_{\lambda \to \mathrm{\infty }}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{v}_{\lambda ,\gamma }{\mathrm{\mid }}^{2}dx=c_0{\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\chi {\mathrm{\mid }}^{2}dS>0$. Together with the upper bound in step 4.1 this proves the two-sided bound for large $\lambda$. On each remaining compact parameter interval the quasimodes vary continuously in $H^1$ and have nonzero boundary value ${\lambda }^{-1\mathrm{/}2}{e}^{i\lambda \varphi }\chi$, so their norms have a positive minimum; enlarge the constants to cover all $\lambda \ge 1$.

The exact corrections satisfy $H^1$ errors $O({\lambda }^{-N})$ by step 5.1, while both approximate beams have uniformly bounded $H^1$ norms by step 4.1. Cauchy-Schwarz bounds the change in the weighted product integral, after multiplication by ${\lambda }^k$, by $C{\lambda }^k{\lambda }^{-N}$ because $0\le \mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega }{)}^k\le C$ on the bounded domain. Choose $N>k$. The error tends to zero, so the same limit holds for the exact solutions.