[proofplan] We prove the global estimate by combining local $H^2$ estimates on a finite cover of $\overline U$. Interior patches are controlled by the interior $H^2$ estimate for the Laplacian, while boundary patches are flattened by $C^2$ coordinate maps and estimated by tangential difference quotients; the equation then recovers the second normal derivative. A smooth [partition of unity](/page/Partition%20of%20Unity) localizes the solution, and the finitely many local constants combine into a global estimate. Finally, Poincare's inequality and the weak formulation control the lower-order $H^1$ terms by $\|f\|_{L^2(U)}$. [/proofplan]

[step:Control the $H^1$ norm of the weak solution by the data]Throughout the proof, $\mathcal{L}^n$ denotes $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$ restricted to the measurable set under integration. For an [open set](/page/Open%20Set) $A \subset \mathbb{R}^n$, let $C_c^\infty(A)$ denote the space of smooth real-valued functions with compact support in $A$, and let $\mathcal{D}'(A)$ denote the space of distributions on $A$, namely the continuous linear functionals on $C_c^\infty(A)$. Since $u \in H^1_0(U)$ satisfies the weak formulation, we may choose $v = u$ and obtain \begin{align*} \int_U |\nabla u|^2 \, d\mathcal{L}^n(x) = \int_U f u \, d\mathcal{L}^n(x). \end{align*} By the [Hilbert space](/page/Hilbert%20Space) inner-product estimate in $L^2(U)$, \begin{align*} \int_U f u \, d\mathcal{L}^n(x) \leq \|f\|_{L^2(U)} \|u\|_{L^2(U)}. \end{align*} By [Poincare's Inequality with Zero Trace](/theorems/76), which applies because $U$ is bounded and $u \in H^1_0(U)$, there exists a constant $C_P = C_P(U) > 0$ such that \begin{align*} \|u\|_{L^2(U)} \leq C_P \|\nabla u\|_{L^2(U)}. \end{align*} Hence \begin{align*} \|\nabla u\|_{L^2(U)}^2 \leq C_P \|f\|_{L^2(U)} \|\nabla u\|_{L^2(U)}. \end{align*} If $\|\nabla u\|_{L^2(U)} = 0$, the desired estimate is immediate for the $H^1$ norm. Otherwise, dividing by $\|\nabla u\|_{L^2(U)}$ gives \begin{align*} \|\nabla u\|_{L^2(U)} \leq C_P \|f\|_{L^2(U)}. \end{align*} Applying [Poincare's Inequality with Zero Trace](/theorems/76) once more and using the standard definition of the $H^1(U)$ norm, \begin{align*} \|u\|_{H^1(U)} \leq C_P(1 + C_P^2)^{1/2} \|f\|_{L^2(U)}. \end{align*}[/step]

[guided]The first estimate extracts the information already contained in the weak formulation. The weak formulation is \begin{align*} \int_U \nabla u \cdot \nabla v \, d\mathcal{L}^n(x) = \int_U f v \, d\mathcal{L}^n(x) \quad \text{for every } v \in H^1_0(U). \end{align*} Because $u \in H^1_0(U)$, the choice $v = u$ is admissible. Substituting $v = u$ gives the energy identity \begin{align*} \int_U |\nabla u|^2 \, d\mathcal{L}^n(x) = \int_U f u \, d\mathcal{L}^n(x). \end{align*} Now we estimate the right-hand side using the Hilbert space inner-product estimate in $L^2(U)$: \begin{align*} \int_U f u \, d\mathcal{L}^n(x) \leq \|f\|_{L^2(U)} \|u\|_{L^2(U)}. \end{align*} The zero boundary condition is used through [Poincare's Inequality with Zero Trace](/theorems/76). Its hypotheses hold because $U$ is bounded and $u \in H^1_0(U)$, so there exists $C_P = C_P(U) > 0$ such that \begin{align*} \|u\|_{L^2(U)} \leq C_P \|\nabla u\|_{L^2(U)}. \end{align*} Combining these inequalities gives \begin{align*} \|\nabla u\|_{L^2(U)}^2 \leq C_P \|f\|_{L^2(U)} \|\nabla u\|_{L^2(U)}. \end{align*} If $\|\nabla u\|_{L^2(U)} = 0$, then Poincare's inequality gives $\|u\|_{L^2(U)} = 0$, so $u = 0$ in $H^1_0(U)$ and the estimate holds. If $\|\nabla u\|_{L^2(U)} > 0$, we divide by $\|\nabla u\|_{L^2(U)}$ and obtain \begin{align*} \|\nabla u\|_{L^2(U)} \leq C_P \|f\|_{L^2(U)}. \end{align*} A second use of [Poincare's Inequality with Zero Trace](/theorems/76), together with the standard definition \begin{align*} \|u\|_{H^1(U)}^2 = \|u\|_{L^2(U)}^2 + \|\nabla u\|_{L^2(U)}^2, \end{align*} then yields \begin{align*} \|u\|_{H^1(U)} \leq C_P(1 + C_P^2)^{1/2} \|f\|_{L^2(U)}. \end{align*} This is the lower-order estimate that will later absorb all local error terms.[/guided]

[step:Choose a finite interior and boundary cover with cutoffs supported inside estimated subpatches]Because $\overline U$ is compact and $\partial U$ is $C^2$, the [Straightening the Boundary Theorem](/theorems/50) supplies boundary flattening charts near each boundary point. Choose finitely many open sets $V_1,\dots,V_N \subset \mathbb{R}^n$ and smaller open sets $O_1,\dots,O_N \subset \mathbb{R}^n$ such that the $O_k$ are open neighbourhoods of their intersections with $\overline U$, $\overline U \subset \bigcup_{k=1}^N O_k$, $\overline{O_k} \subset V_k$, and each $V_k$ is of one of the following two types. For an interior patch, choose $V_k$ to be a Euclidean ball with $\overline{V_k} \subset U$. For a boundary patch, there are an open set $W_k \subset \mathbb{R}^n$, a $C^2$ diffeomorphism $\Phi_k: W_k \to V_k$, and a radius $r_k > 0$ such that $\Phi_k$ maps the flat half-ball $B(0,r_k) \cap \{y_n > 0\}$ onto $V_k \cap U$ and maps $B(0,r_k) \cap \{y_n = 0\}$ onto $V_k \cap \partial U$. Shrinking $O_k$ if necessary, assume that for every boundary patch there is a radius $\rho_k$ with $0 < \rho_k < r_k$ such that, with \begin{align*} Q_k^\flat := B(0,\rho_k) \cap \{y_n > 0\}, \end{align*} one has $O_k \cap U \subset \Phi_k(Q_k^\flat)$. The set $Q_k^\flat$ is a smaller half-ball touching the flat boundary, not a compactly contained subset of the open half-ball. Choose functions $\eta_k \in C_c^\infty(O_k)$ for $1 \leq k \leq N$ such that $0 \leq \eta_k \leq 1$ and the function $S: \overline U \to (0,\infty)$ defined by $S(x) = \left(\sum_{j=1}^N \eta_j(x)^2\right)^{1/2}$ is positive on $\overline U$. Choose an open neighbourhood $N_U \subset \mathbb{R}^n$ of $\overline U$ on which $S>0$, and shrink the sets $O_k$ so that $\operatorname{supp}\eta_k \cap N_U$ has closure contained in $O_k \cap V_k$. Choose a smooth cutoff $\theta_k \in C_c^\infty(O_k \cap V_k)$ equal to $1$ on a neighbourhood of $\operatorname{supp}\eta_k \cap \overline U$, and define $\zeta_k \in C_c^\infty(V_k)$ by $\zeta_k(x) = \theta_k(x)\eta_k(x)/S(x)$ on $N_U$ and by $0$ outside $N_U \cap O_k$. Then $\operatorname{supp}\zeta_k \subset O_k$, $0 \leq \zeta_k \leq 1$ on $V_k$, and \begin{align*} \sum_{k=1}^N \zeta_k^2 = 1 \quad \text{on } \overline U. \end{align*} For each $k$, define the localized function $u_k: U \to \mathbb{R}$ by $u_k(x) = \zeta_k(x) u(x)$. Then $u_k \in H^1_0(U \cap V_k)$, and every point of $\overline U$ lies in some region where the corresponding local estimate is valid because $\operatorname{supp}\zeta_k \subset O_k$ and $O_k$ lies inside the corresponding interior ball or flattened boundary half-ball.[/step]

[guided]The cover must be chosen so that each local estimate is applied on a set where its hypotheses are true. Since $\overline U$ is compact and $\partial U$ is $C^2$, the [Straightening the Boundary Theorem](/theorems/50) gives boundary flattening charts near every boundary point. We choose finitely many open sets $V_1,\dots,V_N \subset \mathbb{R}^n$ and smaller open sets $O_1,\dots,O_N \subset \mathbb{R}^n$ such that $\overline U \subset \bigcup_{k=1}^N O_k$ and $\overline{O_k} \subset V_k$ for every $k$. For an interior patch, $V_k$ is chosen to be a Euclidean ball with $\overline{V_k} \subset U$. For a boundary patch, the flattening theorem gives an open set $W_k \subset \mathbb{R}^n$, a $C^2$ diffeomorphism \begin{align*} \Phi_k: W_k \to V_k, \end{align*} and a radius $r_k > 0$ such that $\Phi_k$ maps $B(0,r_k) \cap \{y_n > 0\}$ onto $V_k \cap U$ and maps $B(0,r_k) \cap \{y_n = 0\}$ onto $V_k \cap \partial U$. Shrinking $O_k$ if necessary, choose $\rho_k$ with $0 < \rho_k < r_k$ and define \begin{align*} Q_k^\flat := B(0,\rho_k) \cap \{y_n > 0\}. \end{align*} Then $O_k \cap U \subset \Phi_k(Q_k^\flat)$. The set $Q_k^\flat$ must still touch the flat boundary $\{y_n = 0\}$; a compactly contained subset of the open half-ball would stay a positive distance from the boundary and would not cover points of $U$ arbitrarily close to $\partial U$. Next choose $\eta_k \in C_c^\infty(O_k)$ for $1 \leq k \leq N$ with $0 \leq \eta_k \leq 1$ and define \begin{align*} S: \overline U \to (0,\infty), \quad S(x) = \left(\sum_{j=1}^N \eta_j(x)^2\right)^{1/2}. \end{align*} The functions $\eta_k$ are chosen so that $S$ is positive on $\overline U$. Choose an open neighbourhood $N_U$ of $\overline U$ on which $S>0$, shrink supports inside $O_k \cap V_k$, and choose $\theta_k \in C_c^\infty(O_k \cap V_k)$ equal to $1$ near $\operatorname{supp}\eta_k \cap \overline U$. Define \begin{align*} \zeta_k(x) = \theta_k(x)\eta_k(x)/S(x) \end{align*} on $N_U$, and extend by $0$ outside $N_U \cap O_k$ inside $V_k$. This construction gives $\zeta_k \in C_c^\infty(V_k)$, $\operatorname{supp}\zeta_k \subset O_k$, $0 \leq \zeta_k \leq 1$, and \begin{align*} \sum_{k=1}^N \zeta_k^2 = 1 \quad \text{on } \overline U. \end{align*} For each $k$, define \begin{align*} u_k: U \to \mathbb{R}, \quad u_k(x) = \zeta_k(x)u(x). \end{align*} Since $\zeta_k$ is smooth and compactly supported in $V_k$, the weak product rule from [Basic Properties of the Weak Derivative](/theorems/77) gives $u_k \in H^1(U \cap V_k)$; the support condition and the trace condition on $u \in H^1_0(U)$ give $u_k \in H^1_0(U \cap V_k)$. Every point of $\overline U$ lies in a region where the corresponding local estimate is valid because $\operatorname{supp}\zeta_k \subset O_k$ and each $O_k$ lies inside the corresponding interior ball or flattened boundary half-ball.[/guided]

[step:Apply the interior $H^2$ estimate on patches compactly contained in $U$] Let $V_k$ be an interior patch. Since $\operatorname{supp}\zeta_k \subset V_k$ and $\overline{V_k} \subset U$, the localized function $u_k = \zeta_k u$ belongs to $H^1_0(V_k)$ and satisfies \begin{align*} -\Delta u_k = \zeta_k f - 2 \nabla \zeta_k \cdot \nabla u - u \Delta \zeta_k \quad \text{in } V_k \end{align*} in the sense of distributions. The right-hand side belongs to $L^2(V_k)$ because $f \in L^2(U)$, $u \in H^1(U)$, and $\zeta_k \in C_c^\infty(V_k)$. We use the [Interior $H^2$ Regularity Theorem](/theorems/95) in the following form: if $V \subset \mathbb{R}^n$ is a Euclidean ball, $z \in H^1_0(V)$, and $-\Delta z = g$ in $\mathcal{D}'(V)$ with $g \in L^2(V)$, then $z \in H^2(V)$ and \begin{align*} \|z\|_{H^2(V)} \leq C_V\left(\|g\|_{L^2(V)} + \|z\|_{L^2(V)}\right) \end{align*} for a constant $C_V > 0$ depending only on $V$. Its hypotheses apply with $V = V_k$, $z = u_k$, and $g = \zeta_k f - 2 \nabla \zeta_k \cdot \nabla u - u \Delta \zeta_k$, because $u_k \in H^1_0(V_k)$ and the preceding paragraph proves $g \in L^2(V_k)$. Hence there exists a constant $C_k > 0$, depending only on $V_k$, such that \begin{align*} \|u_k\|_{H^2(V_k)} \leq C_k \left(\|\zeta_k f\|_{L^2(V_k)} + 2\|\nabla \zeta_k \cdot \nabla u\|_{L^2(V_k)} + \|u \Delta \zeta_k\|_{L^2(V_k)} + \|u_k\|_{L^2(V_k)}\right). \end{align*} Since the derivatives of $\zeta_k$ are bounded, there is a constant $A_k > 0$ such that \begin{align*} \|u_k\|_{H^2(V_k)} \leq A_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} [/step]

[step:Flatten a boundary patch and estimate tangential second derivatives]Let $V_k$ be a boundary patch, and use the $C^2$ diffeomorphism $\Phi_k: W_k \to V_k$ from the finite cover. Define the flattened half-ball \begin{align*} Q_k := B(0,r_k) \cap \{y_n > 0\}. \end{align*} Define the pulled-back localized function \begin{align*} w_k: Q_k \to \mathbb{R}, \quad w_k(y) = u_k(\Phi_k(y)). \end{align*} Since $u_k \in H^1_0(U \cap V_k)$ and $\Phi_k$ maps the flat boundary $\{y_n = 0\}$ onto $\partial U$, the trace of $w_k$ vanishes on $B(0,r_k) \cap \{y_n = 0\}$. Let $P_k: Q_k \to \mathbb{R}^{n \times n}$ denote the Jacobian matrix field $P_k(y) = J\Phi_k(y)$, and let $J_k: Q_k \to (0,\infty)$ denote $J_k(y) = |\det P_k(y)|$. Define $g_k: V_k \to \mathbb{R}$ by $g_k = \zeta_k f - 2 \nabla \zeta_k \cdot \nabla u - u\Delta \zeta_k$ on $U \cap V_k$ and by zero off $U \cap V_k$ inside $V_k$. By the [Change of Variables Theorem](/theorems/22), the weak equation for $u_k$ transforms under $x = \Phi_k(y)$ into the uniformly elliptic divergence-form equation \begin{align*} -\sum_{i,j=1}^n \partial_{y_i}\left(a_{ij}(y)\partial_{y_j} w_k(y)\right) = F_k(y) \quad \text{in } \mathcal{D}'(Q_k), \end{align*} where \begin{align*} a_{ij}(y) = J_k(y)\sum_{\ell=1}^n (P_k(y)^{-1})_{j\ell}(P_k(y)^{-1})_{i\ell} \end{align*} and $F_k: Q_k \to \mathbb{R}$ is defined by $F_k(y) = J_k(y)g_k(\Phi_k(y))$. Since $\Phi_k$ is $C^2$, $a_{ij} \in C^1(\overline{Q_k})$; the positive lower and upper bounds for $J_k$ and $P_k^{-1}$ give uniform ellipticity. Also $F_k \in L^2(Q_k)$ and \begin{align*} \|F_k\|_{L^2(Q_k)} \leq B_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right) \end{align*} for a constant $B_k > 0$ depending only on $\|J\Phi_k\|_{C^1(Q_k)}$, $\|J\Phi_k^{-1}\|_{C^0(V_k)}$, positive lower and upper bounds for $|\det J\Phi_k|$ on $Q_k$, and the $C^2$ norms of $\zeta_k$ on $V_k$. We use the localized flat half-ball form supplied by the $m=0$ case of the [Higher Boundary Elliptic Regularity Theorem](/theorems/97), whose proof is the tangential difference-quotient estimate for $C^1$ uniformly elliptic divergence-form coefficients. In this form, if $Q = B(0,r) \cap \{y_n > 0\}$, $Q^\flat = B(0,\rho) \cap \{y_n > 0\}$ with $0 < \rho < r$, $w \in H^1(Q)$ has zero trace on $B(0,r) \cap \{y_n = 0\}$, the coefficients $a_{ij} \in C^1(\overline Q)$ are bounded and uniformly elliptic, and \begin{align*} -\sum_{i,j=1}^n \partial_{y_i}\left(a_{ij}\partial_{y_j}w\right)=F \quad \text{in } \mathcal{D}'(Q) \end{align*} with $F \in L^2(Q)$, then $\partial_{y_i}\partial_{y_j}w \in L^2(Q^\flat)$ for $1 \leq i \leq n-1$ and $1 \leq j \leq n$, and \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j}w\|_{L^2(Q^\flat)} \leq C_{Q,Q^\flat}\left(\|F\|_{L^2(Q)} + \|w\|_{H^1(Q)}\right), \end{align*} where $C_{Q,Q^\flat}$ depends only on $Q^\flat$, $Q$, the ellipticity constant, and the $C^1$ coefficient bounds. Choose $\sigma_k$ with $\rho_k<\sigma_k<r_k$ and define \begin{align*} Q_k^\sharp := B(0,\sigma_k) \cap \{y_n > 0\}. \end{align*} The hypotheses apply to $w_k$ on the pair $Q_k^\sharp\subset Q_k$ because the preceding paragraph gives the equation in $\mathcal{D}'(Q_k)$, $F_k \in L^2(Q_k)$, the flattened trace vanishes, and the $C^2$ chart gives $C^1$ uniformly elliptic coefficients. Therefore there exists $D_k > 0$ such that \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j} w_k\|_{L^2(Q_k^\sharp)} \leq D_k \left(\|F_k\|_{L^2(Q_k)} + \|w_k\|_{H^1(Q_k)}\right). \end{align*} Using the bound for $F_k$ and boundedness of composition by the fixed $C^2$ diffeomorphism on $H^1$ over the patch, we obtain \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j} w_k\|_{L^2(Q_k^\sharp)} \leq E_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right), \end{align*} where $E_k > 0$ depends only on $Q_k^\sharp$, $Q_k$, the ellipticity constant, $\|J\Phi_k\|_{C^1(Q_k)}$, $\|J\Phi_k^{-1}\|_{C^0(V_k)}$, the bounds for $|\det J\Phi_k|$, and the cutoff construction. Since $Q_k^\flat\subset Q_k^\sharp$, the same estimate also controls these tangential derivatives on $Q_k^\flat$. We keep this estimate in the flattened $y$-coordinates; no assertion about physical $x$-coordinate second derivatives is made until the missing pure $y_n$ derivative has been recovered.[/step]

[guided]The boundary is the only place where the global estimate is not just an interior estimate. We straighten the boundary so that the geometric condition $u = 0$ on $\partial U$ becomes the simple flat condition $w_k = 0$ on $\{y_n = 0\}$. Let \begin{align*} Q_k := B(0,r_k) \cap \{y_n > 0\}. \end{align*} The boundary chart is a $C^2$ diffeomorphism \begin{align*} \Phi_k: W_k \to V_k \end{align*} mapping $Q_k$ onto $U \cap V_k$ and mapping $B(0,r_k) \cap \{y_n = 0\}$ onto $\partial U \cap V_k$. We define \begin{align*} w_k: Q_k \to \mathbb{R}, \quad w_k(y) = u_k(\Phi_k(y)). \end{align*} Because $u_k$ has zero trace on $\partial U \cap V_k$, the trace of $w_k$ vanishes on the flat boundary. This is the reason tangential difference quotients are admissible: translating in directions $e_1,\dots,e_{n-1}$ preserves the flat boundary plane and therefore preserves the zero trace condition after multiplying by a compactly supported cutoff. Let $P_k: Q_k \to \mathbb{R}^{n \times n}$ be $P_k(y) = J\Phi_k(y)$, and let $J_k: Q_k \to (0,\infty)$ be $J_k(y) = |\det P_k(y)|$. The [Change of Variables Theorem](/theorems/22) transforms the Laplacian into a uniformly elliptic divergence-form operator with coefficients \begin{align*} a_{ij}(y) = J_k(y)\sum_{\ell=1}^n (P_k(y)^{-1})_{j\ell}(P_k(y)^{-1})_{i\ell}. \end{align*} Since $\Phi_k$ is a $C^2$ diffeomorphism and the patch is compactly contained in its coordinate domain, these coefficients are $C^1$ and satisfy a uniform ellipticity bound. If $g_k: V_k \to \mathbb{R}$ is defined by $g_k = \zeta_k f - 2\nabla\zeta_k \cdot \nabla u - u\Delta\zeta_k$ on $U \cap V_k$ and by zero elsewhere in $V_k$, then the pulled-back right-hand side is $F_k: Q_k \to \mathbb{R}$, $F_k(y) = J_k(y)g_k(\Phi_k(y))$. Thus the pulled-back equation has the form \begin{align*} -\sum_{i,j=1}^n \partial_{y_i}\left(a_{ij}(y)\partial_{y_j} w_k(y)\right) = F_k(y) \quad \text{in } \mathcal{D}'(Q_k). \end{align*} The function $F_k$ contains the pulled-back forcing term $\zeta_k f$ and the commutator terms created by differentiating the cutoff $\zeta_k$. Because $f \in L^2(U)$, $u \in H^1(U)$, $\Phi_k$ has bounded derivatives up to order two on the patch, and $\zeta_k$ is smooth with compact support in $V_k$, there is a constant $B_k > 0$ such that \begin{align*} \|F_k\|_{L^2(Q_k)} \leq B_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Here $B_k$ depends only on $\|J\Phi_k\|_{C^1(Q_k)}$, $\|J\Phi_k^{-1}\|_{C^0(V_k)}$, positive lower and upper bounds for $|\det J\Phi_k|$ on $Q_k$, and the $C^2$ norms of $\zeta_k$ on $V_k$. Now we use tangential difference quotients. The difference quotient in direction $e_i$ for $1 \leq i \leq n-1$ is admissible because it moves points parallel to the boundary plane. Choose $\sigma_k$ with $\rho_k<\sigma_k<r_k$ and set $Q_k^\sharp := B(0,\sigma_k)\cap\{y_n>0\}$. The precise estimate is the localized flat half-ball form supplied by the $m=0$ case of the [Higher Boundary Elliptic Regularity Theorem](/theorems/97), proved by the tangential difference-quotient argument for $C^1$ uniformly elliptic divergence-form coefficients: if $w \in H^1(Q_k)$ has zero trace on the flat boundary, the coefficients are $C^1$, bounded, and uniformly elliptic, and the divergence-form right-hand side lies in $L^2(Q_k)$, then for every smaller half-ball, in particular for $Q_k^\sharp$, all derivatives $\partial_{y_i}\partial_{y_j}w$ with $1 \leq i \leq n-1$ and $1 \leq j \leq n$ lie in $L^2(Q_k^\sharp)$ and satisfy an estimate controlled by $\|F\|_{L^2(Q_k)} + \|w\|_{H^1(Q_k)}$, with constant depending only on the two half-balls, ellipticity, and the $C^1$ coefficient bounds. These hypotheses have just been verified for $w_k$, so \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j} w_k\|_{L^2(Q_k^\sharp)} \leq D_k \left(\|F_k\|_{L^2(Q_k)} + \|w_k\|_{H^1(Q_k)}\right). \end{align*} The term $\|w_k\|_{H^1(Q_k)}$ is controlled by $\|u\|_{H^1(U)}$ because composition with a fixed $C^2$ diffeomorphism is a bounded map on $H^1$ over the patch. Thus \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j} w_k\|_{L^2(Q_k^\sharp)} \leq E_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right), \end{align*} where $E_k > 0$ depends only on $Q_k^\sharp$, $Q_k$, the ellipticity constant, the $C^1$ coefficient bounds, the Jacobian determinant bounds for $\Phi_k$, and the cutoff construction. Since $Q_k^\flat\subset Q_k^\sharp$, the estimate is also available on $Q_k^\flat$ when we later combine all second derivatives there. We deliberately remain in flattened coordinates at this stage: an arbitrary boundary chart mixes the physical coordinate directions, so this estimate should not yet be interpreted as control of all $x$-second derivatives except one.[/guided]

[step:Recover the pure normal second derivative from the equation in flattened coordinates]Let $e_n := (0,\dots,0,1) \in \mathbb{R}^n$ denote the last standard basis vector, so $\partial_{y_n}^2$ is the second derivative in the flattened normal direction. Use the intermediate boundary half-ball $Q_k^\sharp = B(0,\sigma_k)\cap\{y_n>0\}$ chosen in the tangential estimate step. Since $a_{ij} \in C^1(\overline{Q_k^\sharp})$, the divergence-form equation implies in $\mathcal{D}'(Q_k^\sharp)$ that \begin{align*} -\sum_{i,j=1}^n a_{ij}\partial_{y_i}\partial_{y_j}w_k = F_k + \sum_{i,j=1}^n (\partial_{y_i}a_{ij})\partial_{y_j}w_k. \end{align*} The right-hand side lies in $L^2(Q_k^\sharp)$ because $F_k \in L^2(Q_k)$, $\partial_{y_i}a_{ij} \in L^\infty(Q_k^\sharp)$, and $w_k \in H^1(Q_k)$. Separate the term with $(i,j)=(n,n)$ in this distributional identity: \begin{align*} -a_{nn}\partial_{y_n}^2w_k = F_k + \sum_{i,j=1}^n (\partial_{y_i}a_{ij})\partial_{y_j}w_k + \sum_{(i,j)\neq(n,n)} a_{ij}\partial_{y_i}\partial_{y_j}w_k. \end{align*} Every term on the right belongs to $L^2(Q_k^\flat)$: the first two terms were just identified as $L^2$ on the larger set $Q_k^\sharp$, and the final sum is controlled by the tangential estimate because each pair $(i,j)\neq(n,n)$ has at least one tangential index and weak mixed derivatives commute in distributions. Uniform ellipticity gives $a_{nn}(y) \geq \lambda_k > 0$ on $Q_k^\sharp$ for a constant $\lambda_k$ depending only on the flattened coefficients, so multiplication by $a_{nn}^{-1} \in L^\infty(Q_k^\sharp)$ is allowed. Hence $\partial_{y_n}^2w_k \in L^2(Q_k^\flat)$ and \begin{align*} \|\partial_{y_n}^2 w_k\|_{L^2(Q_k^\flat)} \leq G_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right) \end{align*} for a constant $G_k > 0$ depending only on $Q_k^\flat$, $Q_k^\sharp$, the ellipticity lower bound, the $C^1$ coefficient bounds, the Jacobian determinant bounds for $\Phi_k$, and the cutoff construction. Combining this normal estimate with the tangential estimates gives \begin{align*} \|w_k\|_{H^2(Q_k^\flat)} \leq G_k' \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Finally, composition with the fixed $C^2$ diffeomorphism $\Phi_k^{-1}: V_k \to W_k$ transfers the full flattened $H^2$ bound to the physical patch. Since $O_k \cap U \subset \Phi_k(Q_k^\flat)$, differentiating the composition twice expresses each weak $x$-derivative of $u_k$ as a finite sum of weak $y$-derivatives of $w_k$ of order at most two, with coefficients bounded by the $C^1$ norms of $J\Phi_k$ and $J\Phi_k^{-1}$. Using the [Change of Variables Theorem](/theorems/22) to compare the $L^2$ norms gives \begin{align*} \|u_k\|_{H^2(U \cap O_k)} \leq H_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right), \end{align*} where $H_k > 0$ depends only on $G_k'$, $\|J\Phi_k\|_{C^1(Q_k^\flat)}$, $\|J\Phi_k^{-1}\|_{C^1(V_k)}$, and positive lower and upper bounds for $|\det J\Phi_k|$ on $Q_k^\flat$.[/step]

[guided]After the tangential estimate, the only missing second derivative in flattened coordinates is $\partial_{y_n}^2 w_k$. We keep the same smaller half-ball \begin{align*} Q_k^\flat := B(0,\rho_k) \cap \{y_n > 0\} \end{align*} and choose an intermediate half-ball \begin{align*} Q_k^\sharp := B(0,\sigma_k) \cap \{y_n > 0\} \end{align*} with $\rho_k < \sigma_k < r_k$. The intermediate set is used so that all estimates for derivatives on $Q_k^\flat$ are obtained from an equation valid on the larger set $Q_k^\sharp$. Because $a_{ij} \in C^1(\overline{Q_k^\sharp})$, the product rule for distributions expands the divergence-form equation on $Q_k^\sharp$ as \begin{align*} -\sum_{i,j=1}^n a_{ij}\partial_{y_i}\partial_{y_j}w_k = F_k + \sum_{i,j=1}^n (\partial_{y_i}a_{ij})\partial_{y_j}w_k. \end{align*} The term $F_k$ lies in $L^2(Q_k^\sharp)$ because $Q_k^\sharp \subset Q_k$ and $F_k \in L^2(Q_k)$. The coefficient derivatives satisfy $\partial_{y_i}a_{ij} \in L^\infty(Q_k^\sharp)$, and $w_k \in H^1(Q_k)$, so each product $(\partial_{y_i}a_{ij})\partial_{y_j}w_k$ belongs to $L^2(Q_k^\sharp)$. Hence \begin{align*} F_k + \sum_{i,j=1}^n (\partial_{y_i}a_{ij})\partial_{y_j}w_k \in L^2(Q_k^\sharp). \end{align*} Now isolate the coefficient of the missing pure normal derivative: \begin{align*} -a_{nn}\partial_{y_n}^2w_k = F_k + \sum_{i,j=1}^n (\partial_{y_i}a_{ij})\partial_{y_j}w_k + \sum_{(i,j)\neq(n,n)} a_{ij}\partial_{y_i}\partial_{y_j}w_k. \end{align*} The first two terms on the right are in $L^2(Q_k^\flat)$ because they are in $L^2(Q_k^\sharp)$ and $Q_k^\flat \subset Q_k^\sharp$. For the final sum, take any pair $(i,j) \neq (n,n)$. If $i \leq n-1$, then $\partial_{y_i}\partial_{y_j}w_k$ is one of the derivatives controlled by the tangential estimate on $Q_k^\sharp$. If $i=n$ and $j\leq n-1$, weak mixed derivatives commute in distributions, so $\partial_{y_n}\partial_{y_j}w_k = \partial_{y_j}\partial_{y_n}w_k$, again with a tangential first index and again controlled on $Q_k^\sharp$. Since $a_{ij} \in L^\infty(Q_k^\flat)$, every summand $a_{ij}\partial_{y_i}\partial_{y_j}w_k$ with $(i,j)\neq(n,n)$ belongs to $L^2(Q_k^\flat)$. Uniform ellipticity gives a constant $\lambda_k > 0$ such that \begin{align*} a_{nn}(y) \geq \lambda_k \quad \text{for every } y \in Q_k^\sharp. \end{align*} Therefore $a_{nn}^{-1} \in L^\infty(Q_k^\sharp)$ and multiplication by $a_{nn}^{-1}$ is a bounded operation on $L^2(Q_k^\flat)$. Applying this to the isolated identity gives \begin{align*} \partial_{y_n}^2w_k \in L^2(Q_k^\flat) \end{align*} and \begin{align*} \|\partial_{y_n}^2 w_k\|_{L^2(Q_k^\flat)} \leq G_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Here $G_k > 0$ depends only on $Q_k^\flat$, $Q_k^\sharp$, the ellipticity lower bound, the $C^1$ coefficient bounds, the Jacobian determinant bounds for $\Phi_k$, and the cutoff construction. The tangential estimate already gives \begin{align*} \sum_{i=1}^{n-1}\sum_{j=1}^n \|\partial_{y_i}\partial_{y_j} w_k\|_{L^2(Q_k^\flat)} \leq E_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Combining this estimate with the pure normal estimate controls every second [weak derivative](/page/Weak%20Derivative) of $w_k$ on $Q_k^\flat$, while $\|w_k\|_{H^1(Q_k)}$ is already controlled by $\|u\|_{H^1(U)}$. Hence \begin{align*} \|w_k\|_{H^2(Q_k^\flat)} \leq G_k' \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Finally, composition with the fixed $C^2$ diffeomorphism $\Phi_k^{-1}: V_k \to W_k$ transfers this flattened estimate to the physical patch. Since $O_k \cap U \subset \Phi_k(Q_k^\flat)$, differentiating the composition twice expresses each weak $x$-derivative of $u_k$ as a finite sum of weak $y$-derivatives of $w_k$ of order at most two, with bounded coefficients coming from the $C^1$ norms of $J\Phi_k$ and $J\Phi_k^{-1}$. The [Change of Variables Theorem](/theorems/22) then compares the corresponding $L^2$ norms and gives \begin{align*} \|u_k\|_{H^2(U \cap O_k)} \leq H_k \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right), \end{align*} where $H_k > 0$ depends only on $G_k'$, $\|J\Phi_k\|_{C^1(Q_k^\flat)}$, $\|J\Phi_k^{-1}\|_{C^1(V_k)}$, and positive lower and upper bounds for $|\det J\Phi_k|$ on $Q_k^\flat$.[/guided]

[step:Sum the cutoff estimates and absorb lower-order terms]For every $x \in \overline U$, the identity $\sum_{k=1}^N \zeta_k(x)^2 = 1$ gives the reconstruction $u = \sum_{k=1}^N \zeta_k u_k$ on $U$, where $u_k = \zeta_k u$. The support condition $\operatorname{supp}\zeta_k \subset O_k$ ensures that each product $\zeta_k u_k$ is differentiated only on the set where the corresponding local estimate is valid. Since the family is finite and the derivatives of the fixed cutoffs are bounded, applying the weak product rule from [Basic Properties of the Weak Derivative](/theorems/77) to $\zeta_k u_k$ gives a constant $L>0$, depending only on the chosen cover and cutoffs, such that \begin{align*} \|u\|_{H^2(U)} \leq L\left(\sum_{k=1}^N \|u_k\|_{H^2(U \cap O_k)} + \|u\|_{H^1(U)}\right). \end{align*} The extra $\|u\|_{H^1(U)}$ term records the lower-order terms produced when differentiating the products $\zeta_k u_k = \zeta_k^2 u$. Combining the interior and boundary estimates, define \begin{align*} M := L\left(1 + \sum_{k=1}^N K_k\right), \end{align*} where $K_k$ denotes the local constant from the corresponding interior or boundary estimate. Then $M>0$ depends only on the finite cover, the cutoffs, and the $C^2$ geometry of $U$, all of which were chosen from $U$. Hence \begin{align*} \|u\|_{H^2(U)} \leq M \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} Using the $H^1$ estimate from the first step, \begin{align*} \|u\|_{H^1(U)} \leq C_P(1 + C_P^2)^{1/2} \|f\|_{L^2(U)}. \end{align*} Therefore \begin{align*} \|u\|_{H^2(U)} \leq M \left(1 + C_P(1 + C_P^2)^{1/2}\right)\|f\|_{L^2(U)}. \end{align*} Define \begin{align*} C := M \left(1 + C_P(1 + C_P^2)^{1/2}\right). \end{align*} Then $C = C(U) > 0$ and \begin{align*} \|u\|_{H^2(U)} \leq C \|f\|_{L^2(U)}. \end{align*} In particular $u \in H^2(U)$, which proves the claimed global $H^2$ regularity and estimate.[/step]

[guided]The local estimates control the pieces $u_k = \zeta_k u$, but the theorem is a statement about $u$ itself. The identity $\sum_{k=1}^N \zeta_k^2 = 1$ on $\overline U$ gives $u = \sum_{k=1}^N \zeta_k u_k$ on $U$. Differentiating the products $\zeta_k u_k$ produces derivatives of $u_k$ up to order two, with coefficients given by bounded derivatives of the fixed cutoffs. The lower-order contributions are controlled by the local $H^2$ norms of $u_k$ and by the already established $H^1$ bound for $u$. Thus there is a constant $M>0$, depending only on the cover, the cutoffs, and the $C^2$ geometry of $U$, such that \begin{align*} \|u\|_{H^2(U)} \leq M \left(\|f\|_{L^2(U)} + \|u\|_{H^1(U)}\right). \end{align*} The first step already proved \begin{align*} \|u\|_{H^1(U)} \leq C_P(1 + C_P^2)^{1/2} \|f\|_{L^2(U)}. \end{align*} Substituting this estimate gives \begin{align*} \|u\|_{H^2(U)} \leq M \left(1 + C_P(1 + C_P^2)^{1/2}\right)\|f\|_{L^2(U)}. \end{align*} Defining $C := M \left(1 + C_P(1 + C_P^2)^{1/2}\right)$ gives $C = C(U) > 0$ and therefore \begin{align*} \|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}. \end{align*} This proves both $u \in H^2(U)$ and the asserted global estimate.[/guided]