EAN: 9783031012877

SYNTHESIS LECTURES ON MATHEMATICS & STATISTICS ...

This book gives a necessary and sufficient condition in terms of the scattering amplitude for a
scatterer to be spherically symmetric. By a scatterer we mean a potential or an obstacle. It
also gives necessary and sufficient conditions for a domain to be a ball if an overdetermined
boundary problem for the Helmholtz equation in this domain is solvable. This includes a proof
of Schiffer's conjecture  the solution to the Pompeiu problem  and other symmetry problems for
partial differential equations. It goes on to study some other symmetry problems related to the
potential theory. Among these is the problem of invisible obstacles. In Chapter 5  it provides
a solution to the Navier-Stokes problem in ³. The author proves that this problem has a unique
global solution if the data are smooth and decaying sufficiently fast. A new a priori estimate
of the solution to the Navier-Stokes problem is also included. Finally  it delivers a solution
to inverse problem of the potential theory without the standard assumptions about
star-shapeness of the homogeneous bodies.

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