Interior decay of Neumann modes and kernel forward continuity

Cover $\overline{{\mathrm{\Omega }}^{\mathrm{\prime }}}$ by finitely many balls $B_{\mathrm{\ell }}$ whose concentric doubled balls $2B_{\mathrm{\ell }}$ have closure in $\mathrm{\Omega }$, and put $V={\bigcup }_{\mathrm{\ell }}2B_{\mathrm{\ell }}$. Let $v\in L^2(V;{\mathbb{R}}^d)$. On the volume-mean-zero $H^1$ space, [F3] solves ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }z\cdot \mathrm{\nabla }\phi ={\int }_{V}v\cdot \mathrm{\nabla }\phi$, with $\mathrm{\parallel }z{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }v{\mathrm{\parallel }}_2$. This solution is harmonic with zero Neumann data in the fixed boundary collar. By [F1], its boundary trace has spectral $H_{\mathrm{\Delta }}^k$ norm at most $C_k\mathrm{\parallel }v{\mathrm{\parallel }}_2$ for every integer $k$. Testing this equation with the background Neumann solution $u_{1,e_j}$, and that solution's equation with $z$, yields $\mathrm{\mid }{\int }_{V}v\cdot \mathrm{\nabla }{u}_{1,e_j}\mathrm{\mid }=\mathrm{\mid }⟨e_j,z{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}⟩\mathrm{\mid }\le C_k{w}_j^{-k\mathrm{/}2}\mathrm{\parallel }v{\mathrm{\parallel }}_2$. Taking the $L^2$ dual supremum proves $\mathrm{\parallel }\mathrm{\nabla }{u}_{1,e_j}{\mathrm{\parallel }}_{L^2(V)}\le C_k{w}_j^{-k\mathrm{/}2}$. No boundary regularity for a negative-order prescribed flux has been assumed in this argument.

The correction $w=u_{\gamma ,e_j}-u_{1,e_j}$ satisfies $\int \gamma \mathrm{\nabla }w\cdot \mathrm{\nabla }\phi ={\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}(1-\gamma )\mathrm{\nabla }{u}_{1,e_j}\cdot \mathrm{\nabla }\phi$. Coercivity and step 1.1 give $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C\mathrm{\parallel }\mathrm{\nabla }{u}_{1,e_j}{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}^{\mathrm{\prime }})}\le C_k{w}_j^{-k\mathrm{/}2}$. Adding the background gradient on $V$ gives $\mathrm{\parallel }\mathrm{\nabla }{u}_{\gamma ,e_j}{\mathrm{\parallel }}_{L^2(V)}\le C_k{w}_j^{-k\mathrm{/}2}$. Apply [F2] on each smooth ball $2B_{\mathrm{\ell }}$, restricted to $B_{\mathrm{\ell }}$, and sum the finitely many bounds to obtain the stated rapid $L^p({\mathrm{\Omega }}^{\mathrm{\prime }})$ decay, choosing $k\ge 2L$. All domains and constants were fixed before choosing $\gamma$ or $j$.

The two weak Neumann equations, tested with the opposite solutions and subtracted, give $b_{ij}=⟨({\mathcal{R}}_{\gamma }-{\mathcal{R}}_{{\gamma }_0})e_j,e_i⟩={\int }_{{\mathrm{\Omega }}^{\mathrm{\prime }}}({\gamma }_0-\gamma )\mathrm{\nabla }{u}_{\gamma ,e_j}\cdot \overline{\mathrm{\nabla }{u}_{{\gamma }_0,e_i}}$. Hölder with $1\mathrm{/}q+2\mathrm{/}p=1$ and step 2.1 imply $\mathrm{\mid }{b}_{ij}\mathrm{\mid }\le C_L\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_q(w_i{w}_j{)}^{-L}$. Alternatively, the two rapid $L^2(V)$ bounds from that step and the coefficient supremum norm imply the identical bound with $\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_{\mathrm{\infty }}$. The conjugates and sign are consistent with the self-adjoint weak energy form; only the absolute value is used.

In the boundary basis the smoothed matrix is $b_{ij}{w}_j^{-r\mathrm{/}2}$. Choose $L$ with $2L>(d-1)\mathrm{/}2$. By [F4], ${\sum }_{i,j}\mathrm{\mid }{b}_{ij}{\mathrm{\mid }}^2{w}_j^{-r}\le C_L^2\mathrm{\parallel }\gamma -{\gamma }_0{\mathrm{\parallel }}_q^2({\sum }_i{w}_i^{-2L})({\sum }_j{w}_j^{-2L-r})<\mathrm{\infty }$, with the asserted bound after taking the square root. A matrix with this sum finite defines its Hilbert–Schmidt operator: finite column sums converge in squared Hilbert–Schmidt norm and coincide with the given bounded operator on every finite spectral sum, hence on the dense span. The same proof uses the supremum-norm version of step 3.1. Extending by zero on constants preserves these norms.