Small-data solvability and third differentiability of the cubic DN map

The Axiom of Choice is assumed by the Sobolev algebra, embedding, and Banach implicit-function suppliers [F4]--[F6].

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is assumed by the Sobolev and elliptic regularity interfaces [F1], [F3].

Given: the compact smooth Riemannian domain, real smooth $q$, integer $s>n\mathrm{/}2+2$, and the fixed finite smooth boundary directions.

Sobolev spaces on a compact Riemannian domain with boundary: the chartwise integer Sobolev spaces are Banach spaces, and $H_0^1(M)$ is the $H^1$ closure of interior compactly supported smooth functions. Since $H^s(M)\hookrightarrow H^1(M)$ continuously, $X_s=H^s(M)\cap H_0^1(M)$ is closed in $H^s(M)$.

Weak solvability of the positive scalar conductivity equation: on each compact connected component with nonempty boundary, $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}\le C_{P,j}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2}$ for zero-trace $w$. Compactness and local connectedness give finitely many components, so the maximum of these constants gives the inequality on $M$.

Smooth-manifold elliptic nonresonant solvability and regularity: once the weak Dirichlet kernel is trivial, $-{\mathrm{\Delta }}_g$ has a unique zero-boundary solution for every forcing $F\in L^2(M)$. For integer $k\ge 0$ and arbitrary $F\in H^k(M)$, that solution belongs to $H^{k+2}(M)$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{k+2}}\le C_k\mathrm{\parallel }F{\mathrm{\parallel }}_{H^k}$; no trace condition is imposed on $F$.

For $n\ge 2$, The Sobolev space $W^{k,p}$ is an algebra above the critical index gives the Banach-algebra bound for integer $s$ with $2s>n$ on bounded smooth chart domains; finite localization gives it on $M$. For $n=1$, the one-dimensional fundamental theorem of calculus gives $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le C\mathrm{\parallel }z{\mathrm{\parallel }}_{H^1}$ on bounded intervals. Apply this to derivatives through order $s-1$ and use the product rule to bound each derivative of $uv$ through order $s$ in $L^2$; this gives the same algebra bound on each chart and hence on $M$.

For $n\ge 2$, Higher-order Sobolev embedding gives $H^s\hookrightarrow C^2(\bar{M})$ when $s>n\mathrm{/}2+2$. For $n=1$, $s\ge 3$; apply the one-dimensional $H^1\hookrightarrow C^0$ estimate to derivatives through order two.

Implicit function theorem for Banach spaces: if a $C^3$ map between Banach spaces vanishes at a point and its derivative in the unknown variable is a bounded isomorphism, its nearby zero set is the graph of a unique $C^3$ map.

Collar neighborhood theorem: boundary data on a compact smooth boundary extend smoothly into a collar by constant normal extension followed by a smooth cutoff and finite patching.

In boundary charts, the $g$-orthogonal complement of $T\mathrm{\partial }M$ is a smooth line bundle, and the outward choice defines a smooth unit normal. Thus $u\mapsto g({\mathrm{\nabla }}_{g}u,{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}){\mathrm{\mid }}_{\mathrm{\partial }M}$ is bounded from $C^1(\bar{M})$ to $C^0(\mathrm{\partial }M)$.