- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:43
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
16
2000
3
Schiffer problem and isoparametric hypersurfaces
Vladimir
Shklover
University of Maryland, COLLEGE PARK, UNITED STATES
The Schi ffer Problem as originally stated for Euclidean spaces ( and later for some symmetric spaces) is the following: Given a bounded connected open set $\Omega$ with a regular boundary and such that the complement of its closure is connected, does the existence of a so lution to the Overdetermined Neumann Problem (N) imply that $\Omega$ is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schi ffer Problem to an ar bitrary Riemannian manifold and also the possibility of replacing the condition on the domain to be a ball by more general condition; to have a homogeneous boundary (i.e., boundary , admitting a transitive group of isometries ). We prove that if $\Omega$ has a homogeneous boundary then (N) and (D) always admit solutions ( in fact, for infi nitely many eigenvalues), but the converse statement is not always true. We show that in a number of spaces (symmetric and non symmetric), many domains such that their boundaries are isoparametric hypersurfaces have eigenfunctions for (N) and (D) but fail the Schi ffer Conjecture or even its generalization. These ideas can be extended to other (essentially more compli cated) overdetermined boundary value problems, including higher or der equations and non linear equations which, in a number of important cases, may also have solutions in domains with isoparamet ric ( and not necessarily homogeneous) boundaries. Also, a number of initial/ boundary value problems for time- dependent equations with some extra boundary conditions have solutions for domains with the above boundaries. If a time- dependent equation is non linear and has blow- up, this blow- up occurs at the same time at all the points on the boundary.
General
529
569
10.4171/RMI/283
http://www.ems-ph.org/doi/10.4171/RMI/283