Rough conductivity Alessandrini identity and constant-quotient norm

Let $v$ be the weak ${\gamma }_2$ solution with trace class $f_1$, available by [F2]. The difference $u_1-v$ belongs to $H_0^1$. The real-valuedness of ${\gamma }_2$ makes $a_2$ Hermitian, so $a_2(u_1-v,u_2)=\overline{a_2(u_2,u_1-v)}=0$. By the representative independence in [F1], $(({\mathrm{\Lambda }}_{{\gamma }_1}-{\mathrm{\Lambda }}_{{\gamma }_2})f_1)(f_2)=a_1(u_1,u_2)-a_2(v,u_2)=a_1(u_1,u_2)-a_2(u_1,u_2)$. Subtraction of these two integrals is precisely the asserted conductivity identity. All integrals are absolutely convergent since the coefficients are bounded and the solution gradients are square integrable.

Set $T={\mathrm{\Lambda }}_{{\gamma }_1}-{\mathrm{\Lambda }}_{{\gamma }_2}$ and $b(f,g)=(Tf)(g)$. By [F2], $b(f+c\mathbf{1},g+c^{\mathrm{\prime }}\mathbf{1})=b(f,g)$ for all constants $c,c^{\mathrm{\prime }}$. If $N=\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathcal{L}(\mathcal{D},{\mathcal{D}}^{\ast })}$, then $\mathrm{\mid }b(f,g)\mathrm{\mid }\le N\mathrm{\parallel }f+c\mathbf{1}{\mathrm{\parallel }}_{\mathcal{D}}\mathrm{\parallel }g+c^{\mathrm{\prime }}\mathbf{1}{\mathrm{\parallel }}_{\mathcal{D}}$. Taking the two infima independently gives $\mathrm{\mid }b(f,g)\mathrm{\mid }\le N\mathrm{\parallel }[f]{\mathrm{\parallel }}_E\mathrm{\parallel }[g]{\mathrm{\parallel }}_E$, including zero quotient norms. Thus the descended form has norm at most $N$. Conversely, the quotient map has norm at most one, so a bound $\mathrm{\mid }b(f,g)\mathrm{\mid }\le N^{\mathrm{\prime }}\mathrm{\parallel }[f]{\mathrm{\parallel }}_E\mathrm{\parallel }[g]{\mathrm{\parallel }}_E$ implies $\mathrm{\mid }b(f,g)\mathrm{\mid }\le N^{\mathrm{\prime }}\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathcal{D}}\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathcal{D}}$. Taking suprema over the two unit balls gives $N\le N^{\mathrm{\prime }}$. The operator norm into a conjugate-linear dual is exactly this supremum of absolute form values by the definition of the dual norm. Hence the two operator norms are equal.

The descended form in step 1.2 is exactly the difference of the two induced DN maps, so its norm is the stated $\delta$. Applying its form bound to step 1.1 proves the first integral bound. Since $u_j$ represents $f_j$, the two successive quotient infima give $\mathrm{\parallel }[f_j]{\mathrm{\parallel }}_E\le \mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{\mathcal{D}}\le \mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{H^1}$. Multiplying these inequalities proves the second bound, with no coefficient-dependent norm conversion factor.