Centered Neumann voltage sequences on the three-dimensional ball

Let $V=\{w\in H^1(\mathrm{\Omega }):{\int }_{S^2}\mathrm{Tr}w=0\}$. This is closed by [F2]. For $w\in V$, write $\bar{w}$ for its volume mean. Poincaré gives $\mathrm{\parallel }w-\bar{w}{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2$, and the trace bound and boundary-mean condition give $\mathrm{\mid }\bar{w}\mathrm{\mid }\mathrm{\mid }{S}^2\mathrm{\mid }\le \mathrm{\mid }{S}^2{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{Tr}(w-\bar{w}){\mathrm{\parallel }}_2\le C\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2$. Thus $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2$ on $V$. The form $A_{\sigma }(w,v)=\int \sigma \mathrm{\nabla }w\cdot \overline{\mathrm{\nabla }v}$ is bounded and coercive there, uniformly for $m<\sigma <M$. By [F3], any functional $v\mapsto -\int H\cdot \overline{\mathrm{\nabla }v}$ with $H\in L^2(\mathrm{\Omega }{)}^3$ has a unique solution in $V$, of $H^1$ norm at most $C\mathrm{\parallel }H{\mathrm{\parallel }}_2$.

The background solution is $u_{1,\mathrm{\ell }k}(r,\omega )=r^{\mathrm{\ell }}{Y}_{\mathrm{\ell }k}(\omega )\mathrm{/}\mathrm{\ell }$. It is a harmonic polynomial with normal derivative $Y_{\mathrm{\ell }k}$ and boundary mean zero. For smooth test functions, the divergence theorem [F4] gives its weak equation. The density [F5] and continuity of the trace pass the identity to every $H^1$ test. The same integration by parts on $B_{\rho }$ gives $\mathrm{\parallel }\mathrm{\nabla }{u}_{1,\mathrm{\ell }k}{\mathrm{\parallel }}_{L^2(B_{\rho })}^2={\rho }^{2\mathrm{\ell }+1}\mathrm{/}\mathrm{\ell }$. In particular $A_{\rho }:={\sum }_{\mathrm{\ell }\ge 1}(2\mathrm{\ell }+1){\rho }^{2\mathrm{\ell }+1}\mathrm{/}\mathrm{\ell }<\mathrm{\infty }$, since $(2\mathrm{\ell }+1)\mathrm{/}\mathrm{\ell }\le 3$.

Apply step 1.1 with $H=a\mathrm{\nabla }{u}_{1,\mathrm{\ell }k}$, supported in $B_{\rho }$, to obtain $w_{a,\mathrm{\ell }k}\in V$ satisfying $A_{\sigma }(w_{a,\mathrm{\ell }k},v)=-\int a\mathrm{\nabla }{u}_{1,\mathrm{\ell }k}\cdot \overline{\mathrm{\nabla }v}$. The sum $u_{1,\mathrm{\ell }k}+w_{a,\mathrm{\ell }k}$ solves the required Neumann equation, first for $v\in V$ and then for all $v\in H^1$ by subtracting its boundary mean. The current annihilates constants. Uniqueness follows by testing the difference with itself and using step 1.1. Summing the energy estimates gives $\sum \mathrm{\parallel }{w}_{a,\mathrm{\ell }k}{\mathrm{\parallel }}_{H^1}^2\le C\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}^2{A}_{\rho }$. The trace bound proves $G(a)\in Y$. Moreover $\sum \mathrm{\parallel }\mathrm{\nabla }{u}_{\sigma ,\mathrm{\ell }k}{\mathrm{\parallel }}_{L^2(B_{\rho })}^2\le 2A_{\rho }+2\sum \mathrm{\parallel }{w}_{a,\mathrm{\ell }k}{\mathrm{\parallel }}_{H^1}^2\le C$, uniformly over $\mathcal{A}$, since $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\le \max \{1-m,M-1\}$.

For $a,b\in \mathcal{A}$, subtract the weak equations to obtain $A_{1+a}(u_{1+a,\mathrm{\ell }k}-u_{1+b,\mathrm{\ell }k},v)=-\int (a-b)\mathrm{\nabla }{u}_{1+b,\mathrm{\ell }k}\cdot \overline{\mathrm{\nabla }v}$. By step 1.1 and the summed interior bound in step 3.1, the sequence of differences has ${\mathrm{\ell }}^2(H^1)$ norm at most $C\mathrm{\parallel }a-b{\mathrm{\parallel }}_{\mathrm{\infty }}$. For $h\in E$, the same estimate constructs the asserted derivative sequence $v_{a,h}$ with norm at most $C\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$. Subtracting its equation from the difference equation for $a+h,a$ shows that the remainder has forcing $-h\mathrm{\nabla }(u_{1+a+h}-u_{1+a})$ and hence ${\mathrm{\ell }}^2(H^1)$ norm at most $C\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}^2$. Taking traces proves Fréchet differentiability. Finally, subtract the derivative equations at $a,b$: $A_{1+a}(v_{a,h}-v_{b,h},w)=-\int h\mathrm{\nabla }(u_{1+a}-u_{1+b})\cdot \overline{\mathrm{\nabla }w}-\int (a-b)\mathrm{\nabla }{v}_{b,h}\cdot \overline{\mathrm{\nabla }w}$. The established estimates bound its sequence norm by $C\mathrm{\parallel }a-b{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$, proving the operator-norm Lipschitz derivative bound and the asserted Lipschitz bound for $G$.

By step 2.1, $\mathrm{Tr}{u}_{1,\mathrm{\ell }k}=Y_{\mathrm{\ell }k}\mathrm{/}\mathrm{\ell }$. Its squared sequence norm is ${\sum }_{\mathrm{\ell }\ge 1}(2\mathrm{\ell }+1)\mathrm{/}{\mathrm{\ell }}^2$, which diverges since each summand is at least $2\mathrm{/}\mathrm{\ell }$. Thus the absolute background voltage sequence is not in $Y$, while every centered difference is in $Y$ by step 3.1. This establishes the affine-data distinction without changing the set of measured currents.