Positive Hölder cubic map is continuously differentiable

Bounded inclusion. Compactness of $\bar{\mathrm{\Omega }}$ and the smooth boundary give a finite cover by interior balls and flattened boundary boxes whose smaller concentric patches still cover $\bar{\mathrm{\Omega }}$. In an interior patch, sufficiently close points can be joined by a segment; in a boundary patch, flattening identifies the domain locally with one side of a graph, and the inverse graph map and its inverse are Lipschitz on the smaller box. Therefore there are $\delta ,C_0>0$ such that any $x,y\in \bar{\mathrm{\Omega }}$ with $\mathrm{\mid }x-y\mathrm{\mid }<\delta$ can be joined in $\bar{\mathrm{\Omega }}$ by a curve of length at most $C_0\mathrm{\mid }x-y\mathrm{\mid }$: take a Lebesgue number for the finite smaller-patch cover and pull back the straight segment in the corresponding ball or half-box. For boundary endpoints, approximate both endpoints by interior points in the same patch and pass to the limit using the continuous extensions of $u$ and its first derivatives. The fundamental theorem of calculus along these curves gives $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le C_0\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }x-y\mathrm{\mid }$ for such pairs. For $\mathrm{\mid }x-y\mathrm{\mid }\ge \delta$, the quotient is at most $2{\delta }^{-\alpha }\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$. Consequently $[u{]}_{0,\alpha }\le C(\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathrm{\infty }})$, so $\mathrm{\parallel }u{\mathrm{\parallel }}_Y\le C\mathrm{\parallel }u{\mathrm{\parallel }}_X$.

The Hölder norm is an algebra norm. For $f,g\in Y$, the identity $f(x)g(x)-f(y)g(y)=f(x)(g(x)-g(y))+g(y)(f(x)-f(y))$ gives $[fg{]}_{0,\alpha }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}[g{]}_{0,\alpha }+\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}[f{]}_{0,\alpha }$, hence $\mathrm{\parallel }fg{\mathrm{\parallel }}_Y\le \mathrm{\parallel }f{\mathrm{\parallel }}_Y\mathrm{\parallel }g{\mathrm{\parallel }}_Y$. By [F4], the positive minimum $m={\min }_{\bar{\mathrm{\Omega }}}(1+u)$ exists; if $\mathrm{\parallel }h{\mathrm{\parallel }}_Y<m\mathrm{/}2$, then $1+u+h\ge m\mathrm{/}2>0$, so $\mathcal{U}$ is open.

Hölder composition and remainder. On a sufficiently small $Y$-neighborhood of a fixed $u\in \mathcal{U}$, the values of $1+u$ lie in a compact interval contained in $(0,\mathrm{\infty })$. For $j=3,4,5$, the scalar derivative of $s\mapsto s^{-j}$ is bounded on that interval; the real mean-value theorem [F5] gives $[(1+u{)}^{-j}{]}_{0,\alpha }\le C[u{]}_{0,\alpha }$, and these compositions lie in $Y$. Put $z=1+u$ and $r=h\mathrm{/}z$. Since $z\ge m>0$, $\mathrm{\mid }z(x{)}^{-1}-z(y{)}^{-1}\mathrm{\mid }\le m^{-2}\mathrm{\mid }z(x)-z(y)\mathrm{\mid }$, so $z^{-1}\in Y$ and step 1.2 gives $\mathrm{\parallel }r{\mathrm{\parallel }}_Y\le C\mathrm{\parallel }h{\mathrm{\parallel }}_Y$. Direct algebra gives $(z+h{)}^{-3}-z^{-3}+3z^{-4}h=z^{-3}(6r^2+8r^3+3r^4)\mathrm{/}(1+r{)}^3$. For small $\mathrm{\parallel }h{\mathrm{\parallel }}_Y$, $z+h\ge m\mathrm{/}2$; the same reciprocal estimate gives $(1+r{)}^{-1}=z\mathrm{/}(z+h)$ bounded in $Y$. The algebra estimate of step 1.2 therefore bounds the norm of this remainder after multiplication by $S$ by $C\mathrm{\parallel }h{\mathrm{\parallel }}_Y^2=o(\mathrm{\parallel }h{\mathrm{\parallel }}_Y)$, proving $D{N}_S(u)[h]=-3S(1+u{)}^{-4}h$.

Continuity of the derivative. For $u,v$ in a smaller neighborhood with $q:=1+u$, $p:=1+v$, and $q,p\ge m\mathrm{/}2$, the factorization $q^{-4}-p^{-4}=(p-q)(p+q)(p^2+q^2)\mathrm{/}(p^4{q}^4)$ and the reciprocal estimates of step 2.1 give $\mathrm{\parallel }(1+u{)}^{-4}-(1+v{)}^{-4}{\mathrm{\parallel }}_Y\le C\mathrm{\parallel }u-v{\mathrm{\parallel }}_Y$. Thus $\mathrm{\parallel }D{N}_S(u)-D{N}_S(v){\mathrm{\parallel }}_{Y\to Y}\le 3\mathrm{\parallel }S{\mathrm{\parallel }}_{Y}C\mathrm{\parallel }u-v{\mathrm{\parallel }}_Y$, proving continuity of $D{N}_S$. By [A1] and [F2], $X$ and $Y$ are Banach, so step 2.1 and the derivative continuity just proved establish that $N_S$ is continuously Fréchet differentiable as a map between Banach spaces; by step 1.1, its restriction along the bounded inclusion $X\hookrightarrow Y$ is also $C^1$.