Improved estimates for the linear Molodensky problem
The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S , proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on S . A similar theorem has already been proved by San...
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| Published in: | Journal of geodesy Vol. 98; no. 5 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01-05-2024
Springer Nature B.V |
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| Abstract | The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid
S
, proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on
S
. A similar theorem has already been proved by Sansò and Venuti (J Geod 82:909–916, 2008). Yet the result basically requires that
S
should have an inclination of less than
60
∘
with respect to the vertical, or better to the radial direction. This constraint could result in a severe regularization for the telluroid specially in mountainous areas. The paper revises the result in an effort to improve the above estimates, essentially showing that the inclination of
S
could go up to
75
∘
. At the same time, the proof is made precise mathematically and hopefully more readable in the geodetic community. |
|---|---|
| AbstractList | The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S, proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on S. A similar theorem has already been proved by Sansò and Venuti (J Geod 82:909–916, 2008). Yet the result basically requires that S should have an inclination of less than 60∘ with respect to the vertical, or better to the radial direction. This constraint could result in a severe regularization for the telluroid specially in mountainous areas. The paper revises the result in an effort to improve the above estimates, essentially showing that the inclination of S could go up to 75∘. At the same time, the proof is made precise mathematically and hopefully more readable in the geodetic community. The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S , proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on S . A similar theorem has already been proved by Sansò and Venuti (J Geod 82:909–916, 2008). Yet the result basically requires that S should have an inclination of less than $$60^\circ $$ 60 ∘ with respect to the vertical, or better to the radial direction. This constraint could result in a severe regularization for the telluroid specially in mountainous areas. The paper revises the result in an effort to improve the above estimates, essentially showing that the inclination of S could go up to $$75^\circ $$ 75 ∘ . At the same time, the proof is made precise mathematically and hopefully more readable in the geodetic community. The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S , proving that a solution exists, is unique and is stable in a space of harmonic functions with square integrable gradient on S . A similar theorem has already been proved by Sansò and Venuti (J Geod 82:909–916, 2008). Yet the result basically requires that S should have an inclination of less than 60 ∘ with respect to the vertical, or better to the radial direction. This constraint could result in a severe regularization for the telluroid specially in mountainous areas. The paper revises the result in an effort to improve the above estimates, essentially showing that the inclination of S could go up to 75 ∘ . At the same time, the proof is made precise mathematically and hopefully more readable in the geodetic community. |
| ArticleNumber | 35 |
| Author | Betti, Barbara Sansò, Fernando |
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| Keywords | Geodetic boundary value problem Regularity of the telluroid Spaces of harmonic functions |
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| References | MoritzHAdvanced physical geodesy1980KarlsruheH. Wichmann Eremeev VF, Yurkina MI, Molodenskii MS (1962) Methods for study of the external gravitational field and figure of the earth. Israel Program for Scientific Translations, Jerusalem Cimmino G (1955) Spazi hilbertiani di funzioni armoniche e questioni connesse. In: Equazioni lineari alle derivate parziali. Roma, UMI SansòFVenutiGTopological vector spaces of harmonic functions and the trace operatorJ Geod20057920322110.1007/s00190-005-0461-2 AxlerSBourdonPRameyWPositive harmonic functions, harmonic function theory. Graduate texts in mathematics1992New YorkSpringer HörmanderLThe boundary problems of physical geodesyArch Ration Mech Anal19766215210.1007/BF00251855 MirandaCPartial differential equations of elliptic type19702BerlinSpringer10.1007/978-3-662-35147-5 Sansò F, Sacerdote F (2008) On the universal solvability of classical boundary value problems of Potential theory: contribution from Geodesy. In: VI Hotine Marussi Symp. on Theoretical and Computational Goedesy, IAG Symposia v. 132, Springer, Berlin Helmert FR (1884) Die Mathematischen und Physikalischen Theorien der höheren Geodäsie, vol 2. Teubner, Leipzig SomiglianaCTeoria generale del campo gravitazionale dell’ellissoide di rotazioneMemorie della Societá Astronomica Italiana19294425 YanushauskasAIThe oblique derivative problem of potential theory1989New YorkConsultants Bureau10.1007/978-1-4684-1674-9 FreedenWGerTSchreinerMSpherical functions of mathematical geosciences2009BerlinSpringer10.1007/978-3-540-85112-7 ShlapunovAAOn duality in spaces of solutions to elliptic systems (English version)Sib Math J200243476977710.1023/A:1016344907411 YosidaKFunctional analysis1980BerlingSpringer Krarup T (2006) Letters on Molodensky’s problem. In: Mathematical foundation of geodesy. Springer, Berlin PizzettiPSulla espressione della gravitá alla superficie del geoide, supposto ellissoidicoRend Mat Acc Lincei S189453166172 SansòFSacerdoteFThe scalar boundary value problem of Physical GeodesyManuscr Geodaet1986111528 HeiskanenWAMoritzHPhysical geodesy1967S FranciscoFreeman and Co McLeanWStrongly elliptic systems and boundary integral equations2000CambridgeCambridge University Press GiraudGEquations à integrales principalesAnn. Sc. Ècole Norm. Sup.19345134 SansòFThe geodetic boundary value problem in gravity spaceMemorie Acc Lincei197714S83 SansòFVenutiGOn the explicit determination of stability constants for the linearized geodetic boundary value problemsJ Geod20088290991610.1007/s00190-008-0221-1 MoritzHThe boundary value problem in physical geodesy1965HelsinkyPubl. Isostat. Inst. of IAG Sansò F (1995) The long road from measurements to BVP in Physical Geodesy. Man Geod 20(5):326–344 Stokes GG (1849) On the variation of gravity ou the surface of the Earth. Trans Camb Philos Soc 8 Krarup T (2006). In: Borre K (ed) Mathematical foundation of geodesy. Springer, Berlin Sansò F, Sideris MG (2013). In: Sansò F, Sideris MG (eds) Geoid determination: theory and methods. Springer, Berlin AA Shlapunov (1846_CR23) 2002; 43 1846_CR9 1846_CR7 F Sansò (1846_CR18) 1986; 11 F Sansò (1846_CR22) 2005; 79 C Miranda (1846_CR12) 1970 AI Yanushauskas (1846_CR26) 1989 C Somigliana (1846_CR24) 1929; 4 WA Heiskanen (1846_CR6) 1967 H Moritz (1846_CR14) 1980 1846_CR25 F Sansò (1846_CR17) 1977; 14 K Yosida (1846_CR27) 1980 1846_CR3 1846_CR2 1846_CR20 F Sansò (1846_CR21) 2008; 82 W McLean (1846_CR11) 2000 P Pizzetti (1846_CR15) 1894; 5 1846_CR19 W Freeden (1846_CR4) 2009 S Axler (1846_CR1) 1992 1846_CR16 H Moritz (1846_CR13) 1965 G Giraud (1846_CR5) 1934; 51 1846_CR10 L Hörmander (1846_CR8) 1976; 62 |
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| Snippet | The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid
S
, proving that a solution exists, is... The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S , proving that a solution exists, is... The paper deals with the linearized Molodensky problem, when data are supposed to be square integrable on the telluroid S, proving that a solution exists, is... |
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| Title | Improved estimates for the linear Molodensky problem |
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