Smooth parameter-dependent Dirichlet and compatible Neumann Poisson solvers

Fixed boundary lifts: in the graph coordinates $r=x_3-h(y)$ with the domain on $r<0$ and outward normal $\nu =(-\mathrm{\nabla }h,1)\mathrm{/}\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2}$, take a finite boundary partition and a fixed normal cutoff equal to one near $r=0$; for a localized Dirichlet datum $g_j(y,t)$ set the lift $L_D{g}_j=\rho (r)g_j(y,t)$ with a larger fixed chart cutoff, and for a localized conormal datum ${\kappa }_j$ set $L_N{\kappa }_j=r\rho (r){\kappa }_j(y,t)\mathrm{/}(\epsilon \sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(y){\mathrm{\mid }}^2})$; at $r=0$ the coefficient terms with a differentiated factor vanish because of the factor $r$ while ${\mathrm{\partial }}_{\nu }r=\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h{\mathrm{\mid }}^2}$, so $\epsilon {\mathrm{\partial }}_{\nu }{L}_N{\kappa }_j={\kappa }_j$, and summing the finite chart pieces gives lifts with the exact datum that are linear, time-independent in their spatial profile, smooth on a spatial collar $\times J$ and bounded in $H^{k+2}$ by finite $C^{k+2}$ boundary seminorms.

Dirichlet branch: subtract the lift $L_D{g}_D$ and apply the weak Dirichlet solver and the exact higher-order Dirichlet boundary estimate to the source $f+\epsilon \mathrm{\Delta }{L}_D{g}_D$; the solution is unique and, by the regularity estimate iterated over $m$, lies in $H^m$ for every $m$ on compact time subintervals, hence has a smooth representative on the closed spatial domain times those subintervals; the estimate is local in time because all data seminorms are taken on compact subintervals.

Neumann weak existence: smooth boundary $\kappa$ defines a bounded boundary trace functional by Cauchy-Schwarz and the $H^1\to L^2(\mathrm{\partial }\mathrm{\Omega })$ trace theorem, and the weak Neumann lemma gives existence exactly under the componentwise compatibility condition, with uniqueness modulo one spatial constant per component at each time; the mean-zero coercivity on each component gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }\kappa {\mathrm{\parallel }}_{L^2(\mathrm{\partial }\mathrm{\Omega })})$ after normalisation, and a bounded smooth graph domain has only finitely many components because a finite cover by connected one-sided graph patches meets only finitely many and every component meets one.

Natural-boundary $H^2$ estimate: after subtracting the fixed lift $H=L_N\kappa$, the residual $w=u-H$ has homogeneous Neumann flux and source $f+\epsilon \mathrm{\Delta }H$, whose residual bulk integral vanishes under the original compatibility; flattening and localization change the datum to $\eta p-A\mathrm{\nabla }z\cdot \mathrm{\nabla }\eta +G\cdot \mathrm{\nabla }\eta$ with $p\in L^2$ and $G\in H^1$, the exact terms accounting for the natural flux, and the tangential difference-quotient test $v=-{\delta }_{-h}^k({\eta }^2{\delta }_h^{k}z)$ — a supported $H^1$ test that need not vanish at the boundary — gives by [F2] and [F3] the uniform bound $\mathrm{\parallel }\eta \mathrm{\nabla }{\delta }_h^{k}z{\mathrm{\parallel }}_2\le C(\mathrm{\parallel }p{\mathrm{\parallel }}_2+\mathrm{\parallel }G{\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }z{\mathrm{\parallel }}_{H^1})$; passing to the limit and using Riesz representation supplies all second derivatives with a tangential factor, and the strong equation $-\mathrm{div}(A\mathrm{\nabla }z)=p-\mathrm{div}G$ together with $A_{33}\ge \alpha >0$ recovers the remaining normal derivative, giving the natural-boundary $H^2$ estimate.

Higher orders: if $w\in H^{k+1}$ with homogeneous Neumann data and datum in $H^k$, localize; the localized scalar datum lies in $H^k$ and the vector datum in $H^{k+1}$, tangential differentiation by a multi-index $\alpha$ of order $k$ keeps the principal matrix with scalar datum $D_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\alpha }p\in L^2$ and vector datum $D_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\alpha }G-{\sum }_{0<\beta \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)(D_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\beta }A)\mathrm{\nabla }{D}_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\alpha -\beta }z\in H^1$, since every commutator carries a coefficient derivative and uses derivatives of $z$ through order at most $k+1$; the tangentially differentiated flux need not vanish but is represented by these vector terms in the natural weak identity, to which step 3.1 applies; this controls order-$k+2$ derivatives with at most two normal factors, and expanding the strong equation on each half-patch and recovering successively larger normal counts through the sole $A_{33}$ principal term, with compact test supports lying in the half-patch interior, yields by the finite cover $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}\le C_k(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^k}+\mathrm{\parallel }\kappa {\mathrm{\parallel }}_{C^{k+2}(\mathrm{\partial }\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1})$; tangential tests are justified by the half-space extension and compactly supported smooth density of [F2] without falsely using $H_0^1$ density.

Strong equation and exact Neumann trace: after the estimates give all spatial orders, Sobolev embedding produces a $C^2$ representative on the closure; testing the weak equation against compact interior smooth functions makes the continuous residual $-\epsilon \mathrm{\Delta }u-f$ vanish pointwise by a nonnegative interior bump, and applying the first Green identity to the smooth representative and tests obtained from the fixed smooth Dirichlet lift gives ${\int }_{\mathrm{\partial }\mathrm{\Omega }}(\epsilon {\mathrm{\partial }}_{\nu }u-\kappa )q=0$ for every smooth boundary test $q$, so the continuous boundary difference vanishes pointwise by a nonnegative chart bump supported where it keeps one sign; the classical conormal equality is therefore recovered from the natural weak formulation rather than presupposed.

Joint smoothness in $(x,t)$ and ambient continuation: differences and time derivatives of compatible data stay in the same compatible linear data space, weak $H^1$ continuity follows from the energy estimate, and applying the higher-order estimate to a difference quotient in time minus the solution for the differentiated data makes the error tend to zero in every $H^m$ on compact time patches, since bulk derivatives converge in $H^k$, boundary derivatives in the finite $C^{k+2}$ seminorms and the $H^1$ error tends to zero; induction gives all time derivatives as solutions for the differentiated data, higher-order Sobolev embedding gives one consistent family of all mixed boundary jets, and the fixed parameter-preserving extension [F4] supplies a single smooth ambient continuation; linearity in the data is clear from the construction.