Matching $C^1$ pieces across a hyperplane have no jump derivative

Given: The Axiom of Countable Choice, $n\ge 2$, the two closed half-boxes, the functions $f_{\pm }$ and their matching traces, and a test function $\phi \in C_c^{\mathrm{\infty }}(Q;\mathbb{C})$.

The only choice assumption declared here is the Axiom of Countable Choice, which says that every countable family of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The weak derivative identity for a first coordinate derivative is ${\int }_{Q}u{\mathrm{\partial }}_j\phi dx=-{\int }_{Q}v\phi dx$ for every test function (Weak derivative of a locally integrable function, Test function space d of an open set). Test functions have compact support in $Q$ and extend by zero to smooth compactly supported functions on ${\mathbb{R}}^n$; their boundary values on $\mathrm{\partial }Q$ vanish. The pairing is complex bilinear, without conjugation (Test function space d of an open set).

Membership in $W^{1,p}$ requires an $L^p$ class for the function and for each weak first derivative; the zero multi-index is the function itself (Integer-order Sobolev spaces and their norms, $C^k$ maps and multi-index derivative notation in Euclidean space). A weak derivative value class is unique almost everywhere under Countable Choice (Uniqueness of a weak derivative as an almost-everywhere class).

The closed half-boxes are compact, and continuous real functions on a compact metric space are bounded; apply this to real and imaginary components and to the continuous derivative extensions and test derivatives. The complex modulus is bounded by the sum of the absolute values of its real and imaginary parts (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

A continuous map has Borel preimages of Borel sets, and the Borel sigma algebra on a subspace is the trace of the ambient Borel sigma algebra (A continuous map has Borel preimages of Borel sets, The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra). The half-boxes are closed and hence Borel (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, The Borel sigma-algebra of a topological space). Consequently finite piecewise gluing on the two open half-boxes and the interface is Borel. Continuous test functions and their derivatives are Borel (Continuous functions on Euclidean spaces are Borel measurable); Borel functions on ${\mathbb{R}}^n$ are Lebesgue measurable under Countable Choice (Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$, Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable). Sums and products of real and complex measurable functions remain measurable by the componentwise arithmetic rules (Arithmetic and lattice operations preserve measurability whenever they are defined, Complex Lp classes and Euclidean test-function conventions).

Under Countable Choice, $Q$ has measure $2^n$ and every box with a degenerate side, including the interface inside a bounded box, is null (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). Lebesgue measure on each Euclidean factor is sigma-finite (Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure).

Real and complex $L^p$ classes are formed from measurable representatives with finite $p$-integral, or an essential bound for $p=\mathrm{\infty }$ (The space $L^p(\mu )$ as the quotient by null functions, The function space ${\mathcal{L}}^p(\mu )$ for $0<p<\mathrm{\infty }$, The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions, Complex Lp classes and Euclidean test-function conventions). For $p>0$, $t\mapsto t^p$ is increasing on $[0,\mathrm{\infty })$: on $(0,\mathrm{\infty })$ this follows from its positive derivative and the mean-value theorem, while at zero it follows from $0^p=0$ and positivity of positive powers (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$, Real powers for positive bases, with the zero-base positive-exponent convention, The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$, Continuity and derivatives of positive-base real powers).

If a measurable function is bounded by $C{\mathbf{1}}_Q$ and ${\lambda }_n(Q)<\mathrm{\infty }$, then its modulus and each finite positive power have finite integral: the majorant is a simple function with integral $C{\lambda }_n(Q)$, and the nonnegative integral is monotone (Integrable real and complex functions, and their integrals, Integral over a measurable subset, The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions, Monotonicity and nonnegative homogeneity of the nonnegative integral).

On a closed interval, if a continuous function $G$ is differentiable except at finitely many interior points and an integrable extension $h$ agrees with $G^{\mathrm{\prime }}$ elsewhere, then ${\int }_a^{b}h=G(b)-G(a)$ (Newton–Leibniz remains valid across finitely many exceptional interior points when the primitive is continuous). Bounded functions continuous except at finitely many points are Riemann integrable (A bounded function on $[a,b]$ that is continuous except at finitely many points is Riemann integrable), and bounded Riemann integrable functions have the same Riemann and Lebesgue integrals under Countable Choice (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral). The product rule holds on each smooth real-valued piece; the complex case is obtained componentwise (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

The Euclidean Lebesgue measure on ${\mathbb{R}}^{n-1}\times \mathbb{R}$ is the completion of the product of the factor Lebesgue measures under Countable Choice (The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures). Tonelli-Fubini applies to integrable functions for that completed product and gives measurable integrable sections outside factor-null sets (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

A coordinate permutation is orthogonal and preserves Euclidean Lebesgue measure (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, Lebesgue measure on ${\mathbb{R}}^n$ is invariant under every orthogonal linear map); integrals of integrable functions are invariant under a measure-preserving map (Measure-preserving transformations and systems, Integral invariance under measure-preserving maps). The complex Lebesgue integral is componentwise and linear on $L^1$ (Complex Lp classes and Euclidean test-function conventions, The Lebesgue integral is linear on $L^1(\mu )$).