A function with vanishing Laplacian has zero boundary flux of its gradient on the unit box

Given: The function $v(x,y,z)=x^2-y^2$, the constant function $u\equiv 1$, and the closed unit box with the six-patch presentation of The closed unit box, with its six faces, is an elementary solid region.

For a finite gluing of elementary solid regions, with $u$ of class $C^1$ and $v$ of class $C^2$ on an open neighbourhood of the union, Green's first identity reads ${\iiint }_E(⟨\mathrm{\nabla }u,\mathrm{\nabla }v⟩+u\mathrm{\Delta }v)={\iint }_{\mathrm{\partial }E}u⟨\mathrm{\nabla }v,n⟩$ (Green's first identity on a glued elementary solid region).

The Laplacian is $\mathrm{\Delta }f=\mathrm{div}\mathrm{\nabla }f$ (The Laplacian of a $C^2$ function and of a $C^2$ vector field).

The closed unit box has the six outward faces of The closed unit box, with its six faces, is an elementary solid region. [ex-the-closed-unit-box-is-an-elementary-solid-region]

The divergence theorem is ${\iiint }_E\mathrm{div}F={\iint }_{\mathrm{\partial }E}⟨F,n⟩$ (The divergence theorem on an elementary solid region).

The gradient is the vector of partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

The divergence is the sum of the coordinate partial derivatives (Divergence and curl of a $C^1$ vector field).

For a bounded Jordan set and an integrable function whose sections are integrable outside a content-zero exceptional set, Jordan Fubini computes the multiple integral by iterated section integrals (Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable).

If $a<b$, $G$ is differentiable on $[a,b]$, and $G^{\mathrm{\prime }}=f$ is integrable there, then ${\int }_a^{b}f=G(b)-G(a)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

$⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

Sums and products differentiate termwise in the usual way (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).