Local energy and zero-boundary closure for the charged constraint problem

Extend $v\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ by zero. For fixed $(x_2,x_3)$, the fundamental theorem of calculus gives $v(x_1,x_2,x_3)={\int }_{-R}^{x_1}{\mathrm{\partial }}_{1}v(s,x_2,x_3)ds$. Cauchy--Schwarz bounds its square by $2R{\int }_{-R}^R\mathrm{\mid }{\mathrm{\partial }}_{1}v{\mathrm{\mid }}^{2}ds$. Integrating over the cube gives $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2\le (2R{)}^2\mathrm{\parallel }{\mathrm{\partial }}_{1}v{\mathrm{\parallel }}_2^2\le (2R{)}^2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2$. Pass to an $H^1$-convergent compactly supported smooth approximation to obtain the same inequality on $H_0^1$.

Suppose $u\in C^1(\bar{\mathrm{\Omega }})$ and $u=0$ on the boundary. Take a finite graph cover and a subordinate partition from [F3], with an additional interior term. In a boundary chart $\mathrm{\Psi }(y^{\mathrm{\prime }},t)=(y^{\mathrm{\prime }},t+\phi (y^{\mathrm{\prime }}))$, $t>0$ is the inward graph coordinate; its Jacobian is one and its differential and inverse differential are bounded on the compact chart. Each localized pullback $z$ is $C^1$, vanishes at $t=0$, and is supported away from artificial chart edges. The identity $z(y^{\mathrm{\prime }},t)={\int }_0^t{\mathrm{\partial }}_{t}z(y^{\mathrm{\prime }},s)ds$ gives $\mathrm{\mid }z\mathrm{\mid }\le Ct$. Choose a smooth ${\eta }_{\delta }(t)$ equal to zero for $t\le \delta$, one for $t\ge 2\delta$, with $\mathrm{\mid }{\eta }_{\delta }^{\mathrm{\prime }}\mathrm{\mid }\le C\mathrm{/}\delta$. The support of $z-z{\eta }_{\delta }$ is a layer of volume $O(\delta )$; both $Dz$ and $z{\eta }_{\delta }^{\mathrm{\prime }}$ are uniformly bounded there. Thus $\mathrm{\parallel }z-z{\eta }_{\delta }{\mathrm{\parallel }}_{H^1}=O({\delta }^{1\mathrm{/}2})$. Transport back and sum the finite partition. The resulting $C^1$ functions have compact support in $\mathrm{\Omega }$, and ordinary mollification at a radius smaller than their distance to the boundary approximates each of them in $H^1$ by $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Hence $u\in H_0^1$.

Conversely, let $u\in C^1(\bar{\mathrm{\Omega }})\cap H_0^1$ and choose $u_j\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging in $H^1$. In a fixed smaller graph chart, for a $C^1$ pullback $w$ and $0<t<h$, the identity $w(y^{\mathrm{\prime }},0)=w(y^{\mathrm{\prime }},t)-{\int }_0^t{\mathrm{\partial }}_{t}w(y^{\mathrm{\prime }},s)ds$ gives $\mathrm{\mid }w(y^{\mathrm{\prime }},0){\mathrm{\mid }}^2\le 2\mathrm{\mid }w(y^{\mathrm{\prime }},t){\mathrm{\mid }}^2+2h{\int }_0^h\mathrm{\mid }{\mathrm{\partial }}_{t}w(y^{\mathrm{\prime }},s){\mathrm{\mid }}^{2}ds$. Average over $t\in (0,h)$ and integrate over a smaller fixed tangential base. Bounded chart Jacobians and differentials give $\mathrm{\parallel }w(\cdot ,0){\mathrm{\parallel }}_{L^2}^2\le C_h\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}^2$. Apply this to the localized pullback of $u-u_j$; its boundary value is the localized boundary value of $u$. The right side tends to zero, so that boundary value vanishes almost everywhere. Continuity makes it vanish everywhere: a nonzero value would persist on an open tangential disk of positive area. The finite cover proves $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$.

Steps 1.2 and 2.1 identify zero classical boundary values with the zero-boundary Sobolev closure for the stated $C^1$ representatives. Together with step 1.1 they establish both claims; no unproved trace-to-classical regularity upgrade has been used.