Complutense University Library

On an oblique boundary value problem related to the Backus problem in Geodesy

Díaz Díaz, Jesús Ildefonso and Díaz Díaz, Gregorio and Otero Juez, Jesus (2006) On an oblique boundary value problem related to the Backus problem in Geodesy. Nonlinear Analysis: Real World Applications, 7 (2). pp. 147-166. ISSN 1468-1218

[img] PDF
Restricted to Repository staff only until 31 December 2020.

207kB

Official URL: http://www.sciencedirect.com/science/article/pii/S1468121805000027

View download statistics for this eprint

==>>> Export to other formats

Abstract

We show the existence and uniqueness of a viscosity solution for an oblique nonlinear problem suggested by the study of the Backus problem on the determination of the external gravitational potential of the Earth from surface measurements of the modulus of the gravity force field.

Item Type:Article
Uncontrolled Keywords:elliptic-equations; harmonic-functions; uniqueness; nonlinear oblique boundary value problem; viscosity solutions; Backus problem; gravity potential
Subjects:Sciences > Mathematics > Differential equations
ID Code:15409
References:

S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, New York/Berlin/Heidelberg, 1992.

G.E. Backus, Application of a non-linear boundary-value problem for the Laplace's equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. XXI (1968) 195–221.

G.E. Backus, Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, J. Geophys. Res. 75 (1970) 6339–6341.

G.E. Backus, Determination of the external geomagnetic field from intensity measurements, Geophys. Res. Lett. 1 (1974).

G. Barles, Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations 106 (1993) 90–106.

A. Bjerhammar, L. Svensson, On the geodetic boundary value problem for a fixed boundary surface—a satellite approach, Bull. Geod. 57 (1983) 382–393.

G. Díaz, J.I. Díaz, J. Otero, Some remarks on the Backus problem in Geodesy, Física de la Tierra:Observaciones Geodésicas y Gravimetria 8 (8) (1996) 179–194.

N. Geffen, A nonstandard nonlinear boundary-value problem for harmonic functions, Quart. Appl. Math. (1983) 289–300.

D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin/Heidelberg/New York, 1983.

E.W. Grafarend, The geoid and the gravimetric boundary value problem, Report No. 18, Department of Geodesy, The Royal Institute of Technology, Stockholm, 1989.

B. Heck, On the non-linear geodetic boundary value problem for a fixed boundary surface, Bull. Geod. 63 (1989) 57–67.

W.A. Heiskanen, H. Moritz, Physical Geodesy, W.H. Freeman and Co., San Francisco, 1967.

L. Hörmander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal. 62 (1976) 1–52.

M.C. Jorge, Local existence of the solution to a nonlinear inverse problem in gravitation, Quart. Appl. Math. XLV (1987) 287–292.

M.C. Jorge, R. Magnanini, Explicit calculation of the solution to Backus' problem with a condition for uniqueness, J. Math. Anal. Appl. 173 (1993) 515–522.

K.R. Koch, A.J. Pope, Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth, Bull. Geod. 46 (1972) 467–476.

G.M. Lieberman, N.S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Am. Math. Soc. 295 (1987) 509–546.

P.L. Lions, Neumann type boundary conditions for Hamilton–Jacobi equations, Duke Math. J. 52 (1985) 793–820.

F.J. Lowes, A. de Santis, B. Duka, A discussion of the uniqueness of a Laplacian potential when given only partial field information on a sphere, Geophys. J. Int. 121 (1995) 579–584.

R. Magnanini, On a (new) nonlinear boundary value problem for the Laplace equation, Technical Report No. 86–7, University of Delaware, 1986.

R. Magnanini, A fully nonlinear boundary value problem for the Laplace equation, Lecture Notes Pure and Applied Mathematics, vol. 109, Marcel Dekker, New York, 1987, pp. 327–330.

R. Magnanini, A fully nonlinear boundary value problem for the Laplace equation in dimension two, Appl. Anal. 39 (1990) 185–192.

N.S. Nadirashvili, On the question of the uniqueness of the solution of the second boundary value problem for second order elliptic equations, Math. USSR Sbornik 50 (1985) 325–341.

N.S. Nadirashvili, On a problem with oblique derivative, Math. USSR Sbornik 55 (1986) 397–414.

J. Otero, On the global solvability of the fixed gravimetric boundary value problem, Proceedings II Hotine-Marussi Symposium on Mathematical Geodesy, 1989, pp. 619–632.

L.E. Payne, P.W. Schaffer, Some nonstandard problems for a class of quasilinear second order elliptic equations, Nonlinear Anal. Theory Methods Appl. 18 (1992) 1003–1014.

L.E. Payne, P.W. Schaffer, Some nonstandard problems for the Poisson equation, Quart. Appl. Math. LI (1993) 81–90.

L.E. Payne, H.F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math. 8 (1958) 551–573.

F. Rellich, Darstellung der Eigenwerte von Δu+λu durch ein Randintegral, Math. Z. 46 (1940) 635–636.

F. Sacerdote, F. Sansò, On the analysis of the fixed-boundary gravimetric boundary-value problem, Proceedings II Hotine-Marussi Symposium on Mathematical Geodesy, 1989, pp. 507–516.

E.M. Stein, G. Weiss, On the theory of harmonic functions of several variables, Acta Math. 103 (1960) 25–62.

V.S. Vladimirov, Equations of Mathematical Physics, Marcel Dekker, New York, 1971.

D.P. Zidarov, Solution and uniqueness of the solution of the inverse potential field problem when the absolute values of the field intensity are known, C. R. Acad. Bulg. Sci. 45 (1992) 31–34.

P.A. Akhmetév, A.V. Khokhlov, Classification of harmonic functions in the exterior of the unit ball, Math. Notes 75 (2) (2004) 166–174.

Deposited On:29 May 2012 09:20
Last Modified:06 Feb 2014 10:24

Repository Staff Only: item control page