Mean-zero weak Neumann problem for uniformly elliptic conductivities

Let $V=\{v\in H^1(\mathrm{\Omega }):{\int }_{\mathrm{\Omega }}vdx=0\}$, a closed Hilbert subspace of $H^1(\mathrm{\Omega })$. By [F2], $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}\le C_0\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$ on $V$, where $C_0=(1+C^2{)}^{1\mathrm{/}2}$. By [F1], the form $a_{\gamma }(u,v)={\int }_{\mathrm{\Omega }}\gamma \mathrm{\nabla }u\cdot \overline{\mathrm{\nabla }v}$ is bounded on $V$ and $a_{\gamma }(v,v)\ge m\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2\ge m{C}_0^{-2}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}^2$. Thus it is bounded and coercive on $V$.

Define ${\mathrm{\ell }}_F(v)=F([\mathrm{Tr}v])$ for $v\in V$. By [F3], the quotient trace norm of $[\mathrm{Tr}v]$ is at most $C_1\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$, so $\mathrm{\mid }{\mathrm{\ell }}_F(v)\mathrm{\mid }\le C_1\mathrm{\parallel }F\mathrm{\parallel }\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$; hence ${\mathrm{\ell }}_F$ is bounded on $V$. Applying [F4] to step 1.1 gives a unique $u_0\in V$ with $a_{\gamma }(u_0,v)={\mathrm{\ell }}_F(v)$ for every $v\in V$.

For arbitrary $v\in H^1(\mathrm{\Omega })$ write $v=v_0+c$ with $v_0=v-v_{\mathrm{\Omega }}\in V$ and $c=v_{\mathrm{\Omega }}$. Gradients are unchanged by adding $c$, and $F$ acts on the trace quotient by constants, so the equation for $u_0$ on $V$ implies it for every $v\in H^1$. If $u_1$ represents another solution class, put $w=(u_1-u_0)-(u_1-u_0{)}_{\mathrm{\Omega }}\in V$. Subtracting the two weak equations gives $a_{\gamma }(w,v)=0$ for all $v\in V$; setting $v=w$ and using coercivity from step 1.1 gives $w=0$. Thus the solution is unique modulo constants.

The estimate in [F4] and step 2.1 give $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^1}\le C_0^2{C}_1{m}^{-1}\mathrm{\parallel }F\mathrm{\parallel }$. The bounded trace map [F3] therefore makes the solution class in $\dot{\mathcal{D}}$ a bounded linear function of $F$. By [F5], the compact smooth boundary has finite positive surface measure. Subtracting the surface mean of $\mathrm{Tr}{u}_0$ selects one representative with boundary mean zero; this constant shift preserves the weak equation and is the only such shift. For mean-zero $f\in L^2(\mathrm{\partial }\mathrm{\Omega })$, Cauchy–Schwarz and the trace bound show $F_f$ is continuous, and its zero mean makes it independent of the representative modulo constants. This proves the final assertion.