The Backus problem with a nonlinearity in a boundary condition


                                   by

                          Elena Cherkaeva (U of U, Math)

             JTB 120, 3:20pm  Monday, January 26, 1998

                                Abstract


The talk deals with a problem of reconstruction of a harmonic inside the 
domain function from given on the boundary length of its gradient. 
This problem arises in geophysics, where the gravitational or magnetic 
vector field should be reconstructed from the measurements of the intensity 
of the field. Backus has shown that the solution of this problem is not 
unique. This non-uniqueness is preserved even in the linearized problem, 
that is a non-classical problem with an oblique derivative; the problem 
violates the Fredholm alternative.
The solvability depends on the behavior of the vector of differentiation 
in the boundary condition, near the line M where the vector is tangent to 
the boundary. For the mentioned geophysical problems, either the solution 
is unique (when the normal derivative does not change the sign, which 
corresponds to the gravitational potential), or there exist infinitely 
many solutions (for magnetic potential, where the normal derivative does 
change the sign on the magnetic equator M). To deal with non-uniqueness, 
the problem is reformulated. The correct formulation requires additional 
data on the singular line M. This permits to estimate the range of the 
solutions of the original problem. 
Numerical methods are developed based on the asymptotic expansion 
technique with several small parameters. For data on irregular grids, 
the regularized solution is constructed using Galerkin method. 




Requests for preprints and reprints to: elena@math.utah.edu
This source can be found at http://www.math.utah.edu/applied-math/