Boundary Schauder applied to tangential quotients gains one derivative

Fix a boundary point and translate its coordinates to $p=0$. Work in a smaller half-ball from [F1]. A fixed smooth bounded subdomain $\mathrm{\Omega }$ with a flat face near $0$ can be constructed as follows. Start with a sufficiently small ball of radius $a$ centered at $a{e}_3$. Its lower hemisphere is $\mathrm{\ell }(x^{\mathrm{\prime }})=a-\sqrt{a^2-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2}$ for $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<a$. Choose $0<2\rho <a\mathrm{/}2$ and the two-dimensional cutoff $\theta (x^{\mathrm{\prime }})=\eta (x^{\mathrm{\prime }}\mathrm{/}\rho )$ from [F5]. Replace that lower graph by $(1-\theta )\mathrm{\ell }$ and leave the upper hemisphere unchanged. This surface is smooth: near $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=a$ it is the original sphere, while on the modified cap it is a smooth graph. Its enclosed domain lies in $x_3>0$, has flat lower face for $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }<\rho$, and its closure lies in the closure of the original half-ball if $a$ is small enough. Choose $r>0$ sufficiently small that $\mathrm{\Omega }\cap B_{4r}=B_{4r}^{+}$ and $4r<\rho$. Take the three-dimensional cutoff $\chi (x)=\eta (x\mathrm{/}(2r))$. It equals one on $B_{2r}$ and vanishes outside $B_{4r}$. Thus $z_h=\chi {\delta }_{h}v$ has zero data on the flat face by [F1] and is identically zero near the remaining boundary. The domain and cutoff are fixed independently of sufficiently small admissible $h$.

We first prove a local interchange rule requiring only one of the two second mixed derivatives. Let $u\in C^1$ near an interior coordinate rectangle, let $a,b$ be distinct coordinate directions, and suppose $H={\mathrm{\partial }}_b{\mathrm{\partial }}_{a}u$ exists and is continuous there. For all sufficiently small oriented $s,h$, two applications of [F6] give $u(x+s{e}_a+h{e}_b)-u(x+h{e}_b)-u(x+s{e}_a)+u(x)={\int }_0^s{\int }_0^{h}H(x+r{e}_a+q{e}_b)dqdr$. Divide by $h$ and let $h\to 0$. The left side tends to ${\mathrm{\partial }}_{b}u(x+s{e}_a)-{\mathrm{\partial }}_{b}u(x)$ because $u\in C^1$. The right side tends to ${\int }_0^{s}H(x+r{e}_a)dr$: its error is bounded by $\mathrm{\mid }s\mathrm{\mid }{\sup }_{r,q}\mathrm{\mid }H(x+r{e}_a+q{e}_b)-H(x+r{e}_a)\mathrm{\mid }$, with $r$ between $0$ and $s$ and $q$ between $0$ and $h$. This supremum tends to zero by continuity on the fixed compact rectangle. Indeed, otherwise a sequence of witnessing pairs has a convergent subsequence by [F10], [F3] and [F4], and its two arguments converge to the same point, contradicting continuity. Divide the resulting identity by $s$ and let $s\to 0$. Continuity of $H$ gives existence of ${\mathrm{\partial }}_a{\mathrm{\partial }}_{b}u(x)$ and equality ${\mathrm{\partial }}_a{\mathrm{\partial }}_{b}u(x)=H(x)$. The argument applies at every interior point.

Write $R_h=a^{ij}({\delta }_{h}v{)}_{ij}$ for the forcing in [F1]. The coordinate product rule [F9] and symmetry of $a^{ij}$ give $a^{ij}(z_h{)}_{ij}=\chi R_h+2a^{ij}{\chi }_i({\delta }_{h}v{)}_j+a^{ij}{\chi }_{ij}{\delta }_{h}v$. The coefficients have fixed $C^{0,\alpha }$ bounds and ellipticity constants on this subdomain by [F1]. Every term on the right is uniformly $C^{0,\alpha }$ by [F1], the fixed smooth cutoff and the product inequality in [F7]. For each nonzero sufficiently small $h$, $z_h$ is already a classical $C^{2,\alpha }$ solution with zero boundary data and uniformly bounded supremum norm. The choice hypotheses follow from [F4]. Apply [F2] with $b=c=0$ and $g=0$ to obtain $\mathrm{\parallel }{z}_h{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}\le C$, with $C$ independent of $h$.

Choose a positive admissible sequence $h_j\to 0$ and set $K=\overline{B_r^{+}}$, a nonempty compact metric space by [F10]. The finitely many fields $D^{\beta }{z}_{h_j}$, $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, are uniformly bounded on $K$. Their top-order fields are uniformly equicontinuous by the common Hölder bound from step 2.1. For $\mathrm{\mid }\beta \mathrm{\mid }\le 1$, join two interior points of $B_r^{+}$ by changing their coordinates one at a time. All intermediate segments lie in $B_{2r}^{+}\subset \mathrm{\Omega }$, and their total length is at most $\sqrt{3}$ times the endpoint distance. Applying [F6] to each segment and using the uniform next-derivative bound gives a common Lipschitz bound; continuity extends it to $K$. By [F4] the choice assumptions hold. Apply [F3] successively to the finitely many families and retain subsequences. This yields a single subsequence for which $D^{\beta }{z}_{h_j}\to Z_{\beta }$ uniformly on $K$ for every $\mathrm{\mid }\beta \mathrm{\mid }\le 2$. For interior $x\in B_r^{+}$, $\chi =1$ and differentiability of $v$ give $Z_0(x)=D_{t}v(x)$. Both sides have continuous extensions, so this identity also holds on the flat face.

For any coordinate segment in $B_r^{+}$ and $\mathrm{\mid }\beta \mathrm{\mid }\le 1$, [F6] gives $D^{\beta }{z}_{h_j}(x+t{e}_i)-D^{\beta }{z}_{h_j}(x)={\int }_0^t{D}^{\beta +e_i}{z}_{h_j}(x+s{e}_i)ds$. Uniform convergence permits passage to the limit: the integral error is at most $\mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }{D}^{\beta +e_i}{z}_{h_j}-Z_{\beta +e_i}{\mathrm{\parallel }}_{\mathrm{\infty }}$. Dividing the limiting identity by $t$ and using continuity of $Z_{\beta +e_i}$ gives ${\mathrm{\partial }}_i{Z}_{\beta }=Z_{\beta +e_i}$ in $B_r^{+}$. These fields already extend continuously to $K$. For $\mathrm{\mid }\beta \mathrm{\mid }=2$, passage to the limit in $\mathrm{\mid }{D}^{\beta }{z}_{h_j}(x)-D^{\beta }{z}_{h_j}(y)\mathrm{\mid }\le C\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$ preserves the Hölder bound on $K$. Hence [F7] gives $D_{t}v\in C^{2,\alpha }(K)$. For each purely tangential multi-index $J$ of length $m-1$, write $J=I+e_t$; [F8] at the already known order $m-1$ identifies $D^{J}w=D_t{D}^{I}w$. Thus every purely tangential field $D^{J}w$ of order $m-1$ is $C^{2,\alpha }$ on $K$.

The only case not immediately ordered as two derivatives of an improved tangential field is an outermost tangential derivative with two normal differentiations among its preceding $m$ derivatives. Write those preceding derivatives, using [F8] only through the known order $m$, as $D_3^2{D}^{I}w$, where $I$ is tangential and $\mathrm{\mid }I\mathrm{\mid }=m-2$. Put $v=D^{I}w$ and $u=D_{3}v$. The given $C^{m,\alpha }$ regularity makes $v\in C^2$ and $u\in C^1$ in the interior. By [F8] for $v$, $D_{t}u=D_3{D}_{t}v$. By step 4.1, $D_{t}v\in C^{2,\alpha }(K)$, so $D_3{D}_{t}u=D_3^2{D}_{t}v$ exists and is continuous, with a Hölder extension to $K$. Apply step 1.2 to this $u$ with $a=t$ and $b=3$. It yields $D_t{D}_3^2{D}^{I}w=D_3^2{D}_t{D}^{I}w=D_3^2{D}^{J}w$, where $J=I+e_t$ is purely tangential of order $m-1$. Thus the previously missing ordering exists and equals a second derivative of the improved field $D^{J}w$, including its continuous Hölder extension.

To exhaust all orderings, let $\gamma$ be a multi-index of length $m+1$ with ${\gamma }_3\le 2$, and consider any ordered derivative with this multiset of coordinate directions. Its first $m$ applied differentiations commute by [F8], so it suffices to treat $D_i{D}^{\beta }w$, where $\mathrm{\mid }\beta \mathrm{\mid }=m$ and $\beta +e_i=\gamma$. Unless $i$ is tangential and ${\beta }_3=2$, the multi-index $\beta$ contains a purely tangential multi-index $J$ of length $m-1$. Write $\beta =J+e_j$ and use the known order-$m$ commutation to obtain $D^{\beta }w=D_j(D^{J}w)$. Step 4.1 supplies $D_i{D}_j(D^{J}w)$ with its continuous Hölder extension. The exceptional case is exactly step 5.1. These fields agree for a fixed $\gamma$: if ${\gamma }_3=2$, all equal $D_3^2{D}^{\gamma -2e_3}w$ by the preceding cases; if ${\gamma }_3=1$, an outer tangential direction $t$ is compared with the outer normal direction using $J=\gamma -e_t-e_3$ and $D_t{D}_3(D^{J}w)=D_3{D}_t(D^{J}w)$ by [F8] on the improved $C^2$ field; if ${\gamma }_3=0$, any two distinct outer directions $i,j$ are compared in the same way using $J=\gamma -e_i-e_j$. In the last two cases all reordered derivatives through order $m$ are already equal by [F8]. Hence every claimed ordering exists and is identified without assuming $w\in C^{m+1}$. There are only finitely many relevant $I,t$, so take a common smaller patch if needed; the identified second derivatives have the continuous Hölder extensions supplied by step 4.1 and [F7]. This proves the stated consequence.