Neumann constants and the Dirichlet boundary cost

Given: Countable Choice, a bounded Lipschitz domain $\mathrm{\Omega }$, a real constant $c$, and the unit disk $B\subset {\mathbb{R}}^2$.

Countable Choice is used for Lebesgue Fubini and the Riemann-to-Lebesgue integral comparison. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Neumann variation is the mass of the derivative inside the open domain; Dirichlet variation is the total variation of the zero extension. (Full-space, Dirichlet, and Neumann TV conventions)

The unit disk indicator belongs to BV and has total variation $2\pi$. (A smooth-curve jump has perimeter total variation)

Distributional derivative acts by the negative-transpose test formula. (Spatial BV and distributional total variation)

Lebesgue measure on ${\mathbb{R}}^n$ is finite on bounded sets under Countable Choice. (Lebesgue measure is a Radon measure on R^n)

Fubini identifies integrals of integrable functions on Euclidean product spaces with iterated integrals. (Fubini's theorem for L^1 functions on a sigma-finite product)

Bounded Riemann-integrable functions on compact intervals have the same Lebesgue integral under Countable Choice. (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral)

A continuous derivative is Riemann integrable and its interval integral is the endpoint difference. (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$)