Exact Helmholtz boundary representation with actual traces

Bounded smooth specialization: apply the second Green identity [F1] on $\mathrm{\Omega }\setminus {\bar{B}}_{\delta }(x)$ to $u$ and $y\mapsto G_k(x-y)$; since $(-\mathrm{\Delta }-k^2)u=f$ and $(\mathrm{\Delta }+k^2)G_k=0$ off the pole, the $k^2$ terms cancel and $-{\int }_{\mathrm{\Omega }\setminus B_{\delta }}{G}_k(x-y)f(y)dy={\int }_{\mathrm{\partial }\mathrm{\Omega }}(G_k{\mathrm{\partial }}_{n}u-u{\mathrm{\partial }}_n{G}_k)+{\int }_{\mathrm{\partial }{B}_{\delta }(x)}(G_k{\mathrm{\partial }}_{n^{\mathrm{\prime }}}u-u{\mathrm{\partial }}_{n^{\mathrm{\prime }}}{G}_k)$; the inner-sphere term tends to $-u(x)$ because $G_k=1\mathrm{/}(4\pi \delta )+O(1)$, ${\mathrm{\partial }}_{n^{\mathrm{\prime }}}{G}_k=1\mathrm{/}(4\pi {\delta }^2)+O(1\mathrm{/}\delta )$ with $n^{\mathrm{\prime }}=-(y-x)\mathrm{/}\delta$, the derivative-test term is $O(\delta )$ and the sphere area is $4\pi {\delta }^2$; the volume integral tends to the absolutely convergent ${\int }_{\mathrm{\Omega }}{G}_k(x-y)f(y)dy$ because $f$ is continuous near $x$ and $G_k$ is locally integrable. This gives the displayed interior formula.

Modal branch: for smooth $g$ set $c_{lm}={\int }_{S^2}g{Y}_{lm}d\mathrm{\Omega }$; by [F6] the operator $\mathrm{\Lambda }$ is symmetric for the integral pairing and annihilates constants, and since each $Y_{lm}$ is the restriction of a degree-$l$ homogeneous harmonic polynomial, Euler's identity and the ambient formula for ${\sum }_j{L}_j^2$ give $\mathrm{\Lambda }{Y}_{lm}=-l(l+1)Y_{lm}$; hence for every integer $p\ge 1$ and $l\ge 1$, $[l(l+1){]}^p{c}_{lm}={\int }_{S^2}(-\mathrm{\Lambda }{)}^{p}g{Y}_{lm}d\mathrm{\Omega }$ and Cauchy-Schwarz [F4] gives $\mathrm{\mid }{c}_{lm}\mathrm{\mid }\le \mathrm{\parallel }{\mathrm{\Lambda }}^{p}g{\mathrm{\parallel }}_2[l(l+1){]}^{-p}$.

Radial majorants: with ${\varphi }_l(r)=h_l(kr)\mathrm{/}{h}_l(k{R}_0)$ and $F_l$ of [F5], $\mathrm{\mid }{h}_l(kr)\mathrm{\mid }\mathrm{/}\mathrm{\mid }{h}_l(k{R}_0)\mathrm{\mid }=(R_0\mathrm{/}r)\sqrt{F_l(kr)\mathrm{/}{F}_l(k{R}_0)}\le R_0\mathrm{/}r$; more precisely the monotonicity of $F_l$ and $F_l(k{R}_0)=k^2{R}_0^2\mathrm{\mid }{h}_l(k{R}_0){\mathrm{\mid }}^2$ give $\mathrm{\mid }{\varphi }_l(r)\mathrm{\mid }\le R_0\mathrm{/}r$ and $\mathrm{\mid }{\varphi }_l^{\mathrm{\prime }}(r)\mathrm{\mid }=\mathrm{\mid }k{h}_l^{\mathrm{\prime }}(kr)\mathrm{\mid }\mathrm{/}\mathrm{\mid }{h}_l(k{R}_0)\mathrm{\mid }\le [(l+1)\mathrm{/}r+k]\mathrm{\mid }{\varphi }_l(r)\mathrm{\mid }$, hence $r\mathrm{\mid }{\varphi }_l^{\mathrm{\prime }}(r)-ik{\varphi }_l(r)\mathrm{\mid }\le R_0[(l+1)\mathrm{/}r+2k]\le l+1+2k{R}_0$ for $r\ge R_0$; for every fixed $l$ the explicit outgoing polynomial gives $r({\varphi }_l^{\mathrm{\prime }}(r)-ik{\varphi }_l(r))\to 0$ as $r\to \mathrm{\infty }$.

Weaker trace branch: let $S_t$ be the inner collar surface of [F2] with inner domain ${\mathrm{\Omega }}_t\ni x$; for small $t$ the punctured Green argument of step 1.1 applies on ${\mathrm{\Omega }}_t$, so $u(x)={\int }_{{\mathrm{\Omega }}_t}{G}_k(x-y)f(y)dy+{\int }_{S_t}(G_k{\mathrm{\partial }}_{n_t}u-u{\mathrm{\partial }}_{n_t}{G}_k)dS$; as $t\downarrow 0$ the traces $u{\mathrm{\mid }}_{S_t}$ and ${\mathrm{\partial }}_{n_t}u{\mathrm{\mid }}_{S_t}$ converge uniformly to the actual continuous traces on $S$ and the surface Jacobians and normals converge uniformly by [F2], so the boundary integrals converge to the displayed ones with the compact chart formula; the volume integrals increase to $\mathrm{\Omega }$ and converge by dominated convergence [F3] using the integrable majorant ${\mathbf{1}}_{\mathrm{\Omega }}\mathrm{\mid }{G}_k(x-\cdot )\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }$.

Exterior branch: for a large ball $B_R$ containing $x$ and the obstacle, apply the same punctured identity on $D_R=(D\cap B_R)\setminus {\bar{B}}_{\delta }(x)$; the inner boundary term tends to $-u(x)$ as in step 1.1 and the obstacle boundary contributes the displayed trace integral, while the outer sphere term is ${\int }_{\mathrm{\mid }y\mathrm{\mid }=R}(v{\mathrm{\partial }}_{R}u-u{\mathrm{\partial }}_{R}v)dS$ with $v=G_k(x-y)$; writing it as $\int [v({\mathrm{\partial }}_{R}u-iku)-u({\mathrm{\partial }}_{R}v-ikv)]dS$, the kernel estimates $v=O(R^{-1})$ and ${\mathrm{\partial }}_{R}v-ikv=O(R^{-2})$ for fixed $x$ combine with the prescribed far conditions to bound the integrand by $o(R^{-2})$ uniformly, so the outer term tends to $0$ after multiplication by the area $4\pi R^2$; absolute convergence of the volume term follows from the stated integrability of $G_k(x-\cdot )f$ on $D$.

With $p=2$ and the addition diagonal $\mathrm{\parallel }{Y}_{lm}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{(2l+1)\mathrm{/}(4\pi )}$ [F4], the weighted sum $W={\sum }_{l,m}(l+1+2k{R}_0)\mathrm{\mid }{c}_{lm}\mathrm{\mid }\mathrm{\parallel }{Y}_{lm}{\mathrm{\parallel }}_{\mathrm{\infty }}$ is finite: the $l=0$ term is finite and for $l\ge 1$ the degree-$l$ contribution is at most a constant times $(2l+1)(l+1+2k{R}_0)\sqrt{(2l+1)\mathrm{/}(4\pi )}[l(l+1){]}^{-2}=O(l^{-3\mathrm{/}2})$, which is summable.

By steps 2.3 and 1.3 the modal series and its first radial derivatives are dominated on $r\ge R_0$ by $W<\mathrm{\infty }$, so they converge absolutely and uniformly there and on every closed exterior annulus; the coordinate FTC [F3] identifies the radial derivative of the sum with the summed derivative, and all higher derivatives converge on compact exterior annuli as follows. For the solid harmonics $P_{lm}(x)=\mathrm{\mid }x{\mathrm{\mid }}^l{Y}_{lm}(x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })$, the addition bound on $Y_{lm}$ and a ball of radius $1\mathrm{/}[2(l+1)]$ about $n\in S^2$ give $\mathrm{\mid }{P}_{lm}\mathrm{\mid }\le e^{1\mathrm{/}2}\sqrt{(2l+1)\mathrm{/}(4\pi )}$ on that ball. Hence [F7] bounds each order-$a$ derivative at $n$ by $C_a(l+1{)}^{a+1\mathrm{/}2}$. Chain rules in a finite sphere atlas give polynomial bounds on every fixed angular derivative. The radial ODE and the first logarithmic-derivative bound of step 1.3 give polynomial bounds in $l$ on every fixed radial derivative on a compact annulus. For any required derivative order choose $p$ in step 1.2 sufficiently large that these polynomial factors, the $(2l+1)$ multiplicity and $[l(l+1){]}^{-p}$ have a summable product. Thus every fixed derivative series converges uniformly, and repeated coordinate FTC identifies those series with the actual Cartesian derivatives. Therefore the modal series defines a $C^{\mathrm{\infty }}$ exterior field; at $r=R_0$ completeness of the harmonics [F4] and continuity give the actual trace $g$, and if the coefficients are recorded in the sphere-normalised basis $Y_{lm}^{R_0}(R_{0}n)=R_0^{-1}{Y}_{lm}(n)$ they become $R_0{c}_{lm}$, giving the same displayed expansion.

Each summand ${\varphi }_l(r)Y_{lm}(n)$ is annihilated by $\mathrm{\Delta }+k^2$: the angular eigenvalue and the radial Helmholtz equation of [F5] give the polar Laplacian identity, and finite combinations of two radial and angular derivatives with smooth coefficients identify $\mathrm{\Delta }$; termwise second derivatives prove the equation for the uniformly convergent limit on each compact exterior set, and uniqueness of limits identifies it as the homogeneous exterior solution with the trace $g$.

Bounds: step 1.3 gives $r\mathrm{\mid }u(rn)\mathrm{\mid }\le R_{0}W$ uniformly; and $r\mathrm{\mid }{\mathrm{\partial }}_{r}u(rn)-iku(rn)\mathrm{\mid }\le {\sum }_{l,m}\mathrm{\mid }{c}_{lm}\mathrm{\mid }\mathrm{\parallel }{Y}_{lm}{\mathrm{\parallel }}_{\mathrm{\infty }}r\mathrm{\mid }{\varphi }_l^{\mathrm{\prime }}(r)-ik{\varphi }_l(r)\mathrm{\mid }\le W$ for all $r\ge R_0$, so the amplitude bound holds; for the Sommerfeld residual, given $ϵ$ choose a finite degree cutoff with tail $<ϵ\mathrm{/}2$ by step 2.3 and then the finitely many remaining terms have fixed finite angular suprema and radial residuals tending to $0$ by step 1.3, giving ${\sup }_{n}r\mathrm{\mid }{\mathrm{\partial }}_{r}u(rn)-iku(rn)\mathrm{\mid }\to 0$.