Uniqueness and smallness gaps in Hörmander's local theory for the nonlinear Molodensky problem

The nonlinear Molodensky problem of physical geodesy asks for the Earth's surface and its external gravitational potential from measured values of the gravity potential and gravity magnitude on that unknown surface — a free-boundary problem for the Laplace equation with an oblique derivative condition. Hörmander (1976) proved local existence for data in a small Hölder $H^{2+ε}$ neighborhood via a Nash–Moser iteration, but uniqueness only in smaller $H^{3+ε}$ neighborhoods, under quantitative smallness conditions on averaged residuals of the telluroid (the reference surface obtained by first mapping the data onto a known normal gravity field) that were never verified for realistic topography; he also warned that straightforward iterative solution loses derivatives. Later literature developed alternative formulations (gravity-space, Legendre-transform), an $L^2$ theory for the linearized scalar problem with explicit inclination constants (up to 75°, checked against real digital terrain data), and a smoothed Nash–Hörmander iteration with boundary elements for model problems. The surviving open core is threefold: uniqueness at the same regularity level as existence within Hörmander's framework, quantitative nonlinear conditions satisfiable by the real unsmoothed Earth, and a provably convergent regularized iteration for realistic global data.

Background

Physical geodesy aims to determine the figure of the Earth and its external gravity field from gravimetric observations made on the Earth's own surface. Molodenskii, Eremeev and Yurkina (1960) formulated the corresponding boundary value problem, now called the nonlinear Molodensky problem: given the gravity potential $W$ and the magnitude $g$ of the gravity vector at every point of the (unknown) physical surface of the Earth, find that surface together with a function $w$ harmonic in the exterior of the surface, vanishing at infinity, such that on the surface the trace of $w$ equals $W$ and the magnitude of its gradient equals $g$ (the known centrifugal potential of the Earth's rotation being split off). Because the surface itself is unknown, this is a free-boundary problem; because gravity acts along a direction that is not normal to the surface, the gradient condition is an oblique derivative condition; and because the linearized operator loses derivatives, classical implicit function theorems do not apply. Hörmander (1976) gave the first rigorous local theory. Writing the unknown surface as a graph over the sphere — the reference surface obtained by mapping the data onto a known normal gravity field is called the telluroid, and the required non-degeneracy of its gravity-gradient Jacobian is called the Marussi condition — he recast the problem as a nonlinear oblique Robin problem (a boundary condition mixing values and a directional derivative) and solved it by a Nash–Moser iteration, an implicit function theorem with smoothing operators that tolerates loss of derivatives. His norms $H^a$ on the sphere are Hölder-type classes; for non-integer $a$ , $H^a$ is the ordinary Hölder class. Two of his results define the state of the art. Theorem 1.5.1 states well-posedness under quantitative smallness of averaged residuals $R_0$ through $R_4$ measuring, among other things, the inclination of the telluroid normal against the radial (isozenithal) direction and the variation of the surface geometry; the paper verifies them only for illustrative small-slope examples (up to about 10°), while real topography such as the Himalaya reaches slopes above 45°. Theorem 3.4.1 states existence of a solution for data in a small $H^{2+ε}$ neighborhood, regularity of the surface embedding in $H^a$ whenever the data are in $H^a$ for non-integer $a > 2+ε$ , but uniqueness only in $H^{3+ε}$ neighborhoods of the reference surface. Hörmander acknowledges this gap and suggests that uniqueness plausibly extends to $H^{2+ε}$ if a Hölder condition on the first differential replaces his second-differential estimate (2.1.5); he also warns that straightforward iterative schemes are impractical because of the loss of derivatives and that a modified, regularized iteration is needed. Subsequent work pursued three routes. Sansò (1976, 1979) transformed the problem to gravity space, a change of coordinates using the potential and the gravity direction as coordinates so that the free boundary becomes a fixed one, and solved the resulting nonlinear equation by Newton's method, obtaining local existence and uniqueness there under weaker data regularity. Witsch and Leis (1985, 1986) used a Legendre transform (exchanging the spatial coordinates with derivatives of the potential, which likewise turns the free boundary into a fixed one) to prove local uniqueness in $C^1$ topology and existence in $C^{1,ε}$ for the free problem of a slowly rotating heavy body. Günther (1987) gave a simplified existence proof. Sacerdote and Sansò (1986) introduced the scalar version in which only the potential is unknown; for the linearized $L^2$ theory of that problem Sansò and Venuti (2008) and Sansò (2014) determined explicit stability constants (telluroid-normal inclination up to about 60°), and Sansò and Betti (2024) improved the admissible inclination to 75° and checked the conditions statistically against a 90 m digital terrain model of the Alps, quantifying how much smoothing real topography requires. Costea, Gimperlein and Stephan (2014) and Banz, Costea, Gimperlein and Stephan (2014) built the regularized iteration Hörmander had called for: a smoothed Nash–Hörmander iteration with heat-equation smoothing and a boundary-element discretization (boundary elements discretize the boundary-integral form of the exterior harmonic problem), with a priori error estimates and model-problem numerics. In parallel, satellite positioning based on GNSS (Global Navigation Satellite Systems) now measures the surface directly, so applied geodesy mostly works with the fixed and linearized boundary value problems (Ilk 2021), while the nonlinear free problem remains the theoretical touchstone of the discipline (Sansò 2018).

Problem Statement

Within Hörmander's (1976) formulation of the nonlinear Molodensky problem — data given as potential $W$ and gravity magnitude $g$ on the unknown surface, telluroid as reference surface, Marussi condition (non-degeneracy of the gravity-gradient Jacobian on the telluroid), Hölder classes $H^a$ on the sphere, local solution theory by Nash–Moser iteration — the open question is whether the theory can be completed to a genuine well-posedness theory at matched, minimal regularity, valid under conditions satisfiable by the real Earth, with a provably convergent solution iteration. Concretely: (1) Uniqueness at existence regularity: prove that any small $H^{2+ε}$ neighborhood of the reference surface embedding contains at most one solution pair (surface embedding, exterior harmonic potential) — extending Theorem 3.4.1(c), which asserts this only for $H^{3+ε}$ neighborhoods — or construct, for a single data pair $(W, g)$ , an explicit pair of distinct solutions whose surface embeddings both lie in a small $H^{2+ε}$ neighborhood, establishing that the gap is genuine. (2) Realistic smallness conditions: determine whether Theorem 1.5.1-type well-posedness, with the averaged residuals $R_0$ through $R_4$ or suitable substitutes, holds in the nonlinear setting for the actual Earth's telluroid — topography that is not smoothed or averaged, with slopes of 45° and beyond — or quantify precisely what smoothing of the data is required for the nonlinear theory to apply. (3) Convergent regularized iteration: establish a provably convergent Nash–Hörmander-type iteration with per-step regularization for realistic global data (e.g. a digital elevation model combined with a global gravity model as starting data), beyond the model problems treated so far. The common objective is a local theory of the nonlinear Molodensky problem in which existence and uniqueness hold in the same low-regularity class, under quantitative geometric conditions verifiable for the real Earth, together with a convergent implementable solver; the components may be resolved separately.

References

L. Hörmander, The boundary problems of physical geodesy, Archive for Rational Mechanics and Analysis 62 (1976), no. 1, pp. 1–52. DOI: 10.1007/BF00251855, https://doi.org/10.1007/BF00251855
M. S. Molodenskii, V. F. Eremeev, M. I. Yurkina, Methods for the study of the external gravitational field and the figure of the Earth (Russian), Transactions of the Central Research Institute of Geodesy, Aerial Photography and Cartography No. 131, Moscow, 1960; English translation: Israel Program for Scientific Translations, Jerusalem, 1962. zbMATH: https://zbmath.org/0117.17704
F. Sansò, Discussion on the existence and uniqueness of the solution of Molodensky's problem in gravity space, Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali (8) 61 (1976), no. 3–4, pp. 260–268. zbMATH: https://zbmath.org/0371.31005 and https://geodesic.mathdoc.fr/item/RLINA_1976_8_61_3-4_a16/
F. Sansò, The gravity space approach to the geodetic boundary value problem including rotational effects, Manuscripta Geodaetica 4 (1979), no. 3, pp. 207–244. DOI: 10.1007/BF03654941, https://doi.org/10.1007/BF03654941
K. J. Witsch, R. Leis, On a free boundary value problem of physical geodesy, I (uniqueness), Mathematical Methods in the Applied Sciences 7 (1985), no. 1, pp. 269–289. DOI: 10.1002/mma.1670070118, https://doi.org/10.1002/mma.1670070118
K. J. Witsch, R. Leis, On a free boundary value problem of physical geodesy, II (existence), Mathematical Methods in the Applied Sciences 8 (1986), no. 1, pp. 1–22. DOI: 10.1002/mma.1670080102, https://doi.org/10.1002/mma.1670080102
M. Günther, Ein einfacher Existenzbeweis für das nichtlineare MOLODENSKY-Problem, Mathematische Nachrichten 130 (1987), no. 1, pp. 251–265. DOI: 10.1002/mana.19871300124, https://doi.org/10.1002/mana.19871300124
F. Sacerdote, F. Sansò, The scalar boundary value problem of physical geodesy, Manuscripta Geodaetica 11 (1986), no. 1, pp. 15–28. DOI: 10.1007/BF03655147, https://doi.org/10.1007/BF03655147
F. Sansò, G. Venuti, On the explicit determination of stability constants for linearized geodetic boundary value problems, Journal of Geodesy 82 (2008), no. 12, pp. 909–916. DOI: 10.1007/s00190-008-0221-1, https://doi.org/10.1007/s00190-008-0221-1
F. Sansò, Geodetic Boundary Value Problem, in: Handbook of Geomathematics, Springer, 2014, pp. 1–27. DOI: 10.1007/978-3-642-27793-1_74-2, https://doi.org/10.1007/978-3-642-27793-1_74-2
F. Sansò, The Analysis of the Geodetic Boundary Value Problem: State and Perspectives, in: Handbook of Mathematical Geodesy, Geosystems Mathematics, Springer, 2018, pp. 459–489. DOI: 10.1007/978-3-319-57181-2_5, https://doi.org/10.1007/978-3-319-57181-2_5
A. Costea, H. Gimperlein, E. P. Stephan, A Nash–Hörmander iteration and boundary elements for the Molodensky problem, Numerische Mathematik 127 (2014), no. 1, pp. 1–34 (published online 2013). DOI: 10.1007/s00211-013-0579-8, https://doi.org/10.1007/s00211-013-0579-8; arXiv:1308.2561
L. Banz, A. Costea, H. Gimperlein, E. P. Stephan, Numerical simulations for the non-linear Molodensky problem, Studia Geophysica et Geodaetica 58 (2014), no. 4, pp. 489–504. DOI: 10.1007/s11200-013-0141-2, https://doi.org/10.1007/s11200-013-0141-2; arXiv:1308.2562
F. Sansò, B. Betti, Improved estimates for the linear Molodensky problem, Journal of Geodesy 98 (2024), no. 5, article 35. DOI: 10.1007/s00190-024-01846-1, https://doi.org/10.1007/s00190-024-01846-1
K. H. Ilk, Das freie Randwertproblem von Molodensky, in: Grundlagen der Physikalischen und Mathematischen Geodäsie, Springer, 2021, pp. 127–168. DOI: 10.1007/978-3-662-62371-8_7, https://doi.org/10.1007/978-3-662-62371-8_7