A uniqueness theorem for a Robin boundary value problem of physical geodesy

Author:
Jesús Otero

Journal:
Quart. Appl. Math. **56** (1998), 245-257

MSC:
Primary 35J25; Secondary 86A30

DOI:
https://doi.org/10.1090/qam/1622562

MathSciNet review:
MR1622562

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We get a uniqueness theorem for a Robin type boundary value problem for the Laplace equation arising in Physical Geodesy in the context of the gravimetric determination of the geoid. The boundary is an oblate ellipsoid of revolution and we have uniqueness of solutions provided that its eccentricity is (approximately) less than 0.526428.

**[1]**A. Bjerhammar,*On the determination of the shape of the geoid and the shape of the earth from an ellipsoidal surface of reference*, Bull. Géodésique (N.S.) No.**81**(1966), 235–265. MR**0200064****[2]**Manfredo P. do Carmo,*Differential geometry of curves and surfaces*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR**0394451****[3]**Eugene Fabes,*Layer potential methods for boundary value problems on Lipschitz domains*, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 55–80. MR**973881**, https://doi.org/10.1007/BFb0103344**[4]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[5]**W. A. Heiskanen and H. Moritz,*Physical Geodesy*, W. H. Freeman and Co., San Francisco/Londres, 1967**[6]**Lars Hörmander,*The boundary problems of physical geodesy*, Arch. Rational Mech. Anal.**62**(1976), no. 1, 1–52. MR**0602181**, https://doi.org/10.1007/BF00251855

L. Hörmander,*Correction to: “The boundary problems of physical geodesy” (Arch. Rational Mech. Anal. 62 (1976), no. 1, 1–52)*, Arch. Ration. Mech. Anal.**65**(1977), no. 4, 395. MR**0602188**, https://doi.org/10.1007/BF00250435**[7]**Lars Hörmander,*Uniqueness theorems and estimates for normally hyperbolic partial differential equations of the second order*, Tolfte Skandinaviska Matematikerkongressen, Lund, Lunds Universitets Matematiska Institution, Lund, 1954, pp. 105–115. MR**0065783****[8]**T. M. MacRobert,*Spherical harmonics. An elementary treatise on harmonic functions with applications*, Third edition revised with the assistance of I. N. Sneddon. International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford-New York-Toronto, Ont., 1967. MR**0220985****[9]**M. S. Molodensky, V. F. Eremeev, and M. I. Yurkina,*Methods for study of the external gravitational field and figure of the earth*, Transl. from Russian (1960), Israel Program for Scientific Translations, Jerusalem, 1962**[10]**H. Moritz,*Advanced Physical Geodesy*, Herbert Wichmann and Abacus Press, Karlsruhe/Tunbridge, 1980**[11]**H. Moritz,*Geodetic reference system*1980, Bull. Geod.**62**, No. 3, 348-358 (1988)**[12]**J. Otero and J. Capdevila,*A series solution for Zagrebin's problem*, Geodetic Theory Today: III Hotine-Marussi Symposium on Mathematical Geodesy (edited and convened by F. Sansò), International Association of Geodesy Symposia, Symposium No. 114, Springer-Verlag, Berlin/Heidelberg/New York, 1995, pp. 280-293**[13]**F. Sansò and G. Sona,*The challenge of computing the geoid in the nineties*, Surveys in Geophysics**14**, 339-371 (1993)**[14]**M. I. Višik,*On an inequality for the boundary values of harmonic functions in a sphere*, Uspehi Matem. Nauk (N.S.)**6**(1951), no. 2(42), 165–166 (Russian). MR**0043279****[15]**V. S. Vladimirov,*Equations of mathematical physics*, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. MR**0268497****[16]**J. Wloka,*Partial differential equations*, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR**895589****[17]**S. Wolfram,*Mathematica: A System for Doing Mathematics by Computer*, 2nd edition, Addison-Wesley Publishing Company, Inc., Redwood City, California, 1991

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35J25,
86A30

Retrieve articles in all journals with MSC: 35J25, 86A30

Additional Information

DOI:
https://doi.org/10.1090/qam/1622562

Article copyright:
© Copyright 1998
American Mathematical Society