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Cubic semilinear DN identification by third linearization

Statement

Assume the Axiom of Choice and Countable Choice.

Let $(M, g)$ be a three-dimensional transversally anisotropic product domain with compact connected simple surface transversal $(M_{0}, g_{0})$ . Let $q_{1}, q_{2} \in C^{\infty} (\overline{M}, R)$ . Suppose that for every finite family of smooth boundary directions, the corresponding small-data nonlinear Dirichlet-to-Neumann maps for $- Δ_{g} u + q_{j} u^{3} = 0$ agree for all sufficiently small real parameter values. Then $q_{1} = q_{2}$ on $\overline{M}$ .

Statement references

Product geometry

Small-data nonlinear DN maps

Third derivative identity

Four-factor concentration

Facts & Assumptions

[A1]

The Axiom of Choice is inherited from the small-data, third-linearization, and four-factor suppliers [F1]--[F3].

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is inherited from the small-data, third-linearization, and four-factor suppliers [F1]--[F3].

Given: the CTA product with simple surface transversal, real smooth cubic coefficients, and equality of the finite-parameter nonlinear DN maps for every finite list of smooth boundary directions.

[F1]

Small-data solvability and third differentiability of the cubic DN map: each finite-parameter solution and its nonlinear DN data are $C^{3}$ in the real parameters near zero.

[F2]

Third linearization identity for the cubic Dirichlet-to-Neumann map: the third DN derivative determines $6 \int_{M} q v_{1} v_{2} v_{3} \overline{v_{4}} d V_{g}$ for any four harmonic boundary solutions.

[F3]

Four-harmonic-product realization of a longitudinal Fourier integral: the claimed four-factor limit at any transversal intersection is a nonzero constant times the longitudinal Fourier transform of the coefficient difference.

[F4]

Uniqueness of the L1 Fourier transform: an $L^{1} (R)$ function with identically zero Fourier transform vanishes almost everywhere.

[F5]

The four-factor supplier imports the smooth harmonic-product density theorem of Lassas--Liimatainen--Lin--Salo, Proposition 5.1, PDF p. 14; its hypotheses are a compact oriented CTA manifold of dimension at least three, a $C^{1}$ coefficient, and four or more smooth harmonic factors. The orientation hypothesis follows here because the local definition Simple surface and geodesic X-ray transform agrees with Paternain--Salo--Uhlmann, Definition 3.8.1: their Theorem 3.8.2 and Propositions 3.8.4--3.8.5 yield unique geodesics and show $M_{0}$ is diffeomorphic to a closed disk, hence orientable; $M$ inherits the product orientation. See Geometric inverse problems, with emphasis on two dimensions, Section 3.8 (author-hosted full text). For this adjudication the complete argument in LL--LS Section 5, PDF pp. 14--23, including Lemma 5.2, and the cited simple-manifold statements and proofs were read; only the density conclusion is imported, not locally reconstructed. The local four-factor supplier uses that density conclusion and rescales one smooth harmonic factor to realize the target integral. Its witnesses are therefore smooth on $\overline{M}$ , depend on the coefficient being tested, and are constant in $τ$ . They are distinct from the $τ$ -dependent Gaussian beams used to prove density in LL--LS. Thus the smooth-boundary identity [F2] applies directly; no extension of that identity to arbitrary $H^{1}$ boundary traces is required. Theorem 1.3, pp. 5--6, is the corresponding nonlinear uniqueness theorem, not a replacement local proof here.

Proof

Given: the data in the statement.

1.1A1A2F1F2givenalgebra

Fix four smooth boundary directions $h_{1}, h_{2}, h_{3}, h_{4}$ . Equality of the finite-parameter DN maps for $q_{1}$ and $q_{2}$ implies equality of their mixed third derivatives at the origin. Applying [F2] to each coefficient and subtracting gives $\int_{M} (q_{1} - q_{2}) v_{1} v_{2} v_{3} \overline{v_{4}} d V_{g} = 0$ for every four harmonic functions $v_{j}$ arising from smooth boundary data.

2.1F2step 1.1algebra

The identity in step 1.1 holds also for complex smooth harmonic functions: write their boundary values as real and imaginary parts and expand the separately linear or conjugate-linear expression. Since the Laplacian has real coefficients, conjugation preserves harmonicity; replacing the fourth function by its conjugate therefore gives $\int_{M} Q w_{1} w_{2} w_{3} w_{4} d V = 0$ for every four smooth complex harmonic functions, where $Q = q_{1} - q_{2}$ .

3.1A1A2F3F5step 2.1

Fix $p \in M_{0}^{\circ}$ and $ξ \in R$ . Choose the transverse boundary-to-boundary geodesic segments through $p$ supplied by [F3] (their existence is established in that supplier's proof). Apply [F3] with $Q = q_{1} - q_{2}$ . Its witnesses are smooth by [F5], so their product integral is zero for every $τ$ by step 2.1. Taking their stated limit gives $c (p, γ_{1}, γ_{2}) \int_{I_{p}} e^{- i ξ x_{1}} (q_{1} - q_{2}) (x_{1}, p) d x_{1} = 0,$ where $c > 0$ is the crossing normalization. The coefficient transform therefore vanishes for every $ξ$ and every $p$ .

4.1F4step 3.1algebra∎

For each $p$ , extend $x_{1} \mapsto (q_{1} - q_{2}) (x_{1}, p)$ by zero outside $I_{p}$ . It is a compactly supported $L^{1} (R)$ function. By step 3.1, [F4] makes this section zero almost everywhere; its smoothness in the interior of $M$ then makes $q_{1} - q_{2} = 0$ on $M^{\circ}$ , and continuity gives equality on $\overline{M}$ . Thus the theorem follows.

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Sources

Lassas, Liimatainen, Lin, and Salo, Inverse problems for elliptic equations with power type nonlinearities (2019 preprint; 2021 publication) (standard reference, not scraped)
Mikko Salo, On geometric inverse problems and microlocal analysis (2023) (standard reference, not scraped)
Feizmohammadi and Oksanen, An inverse problem for a semi-linear elliptic equation in Riemannian geometries (2019) (standard reference, not scraped)