Schiffer problem and isoparametric hypersurfaces.

Shklover, Vladimir E.. "Schiffer problem and isoparametric hypersurfaces.." Revista Matemática Iberoamericana 16.3 (2000): 529-569. <http://eudml.org/doc/39616>.

@article{Shklover2000,
abstract = {The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also the posibility of replacing the condition on the domain to be a ball by [a] more general condition: to have a homogeneous boundary (i.e., boundary admitting a transitive group of isometries). We prove that if Ω has a homogeneous boundary, then (N) and (D) always admit solutions (in fact, for infinitely 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 Schiffer Conjecture or even its generalization.These ideas can be extended to other (essentially more complicated) overdetermined boundary value problems, including higher order equations and non-linear equations, which, in a number of important cases, may also have solutions in domains with isoparametric (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.},
author = {Shklover, Vladimir E.},
journal = {Revista Matemática Iberoamericana},
keywords = {Hipersuperficies; Variedad riemanniana; Variedad simétrica; Problema de contorno; Schiffer problem; isoparametric hypersurfaces; homogeneous boundary value problem; overdetermined boundary value problem},
language = {eng},
number = {3},
pages = {529-569},
title = {Schiffer problem and isoparametric hypersurfaces.},
url = {http://eudml.org/doc/39616},
volume = {16},
year = {2000},
}

TY - JOUR
AU - Shklover, Vladimir E.
TI - Schiffer problem and isoparametric hypersurfaces.
JO - Revista Matemática Iberoamericana
PY - 2000
VL - 16
IS - 3
SP - 529
EP - 569
AB - The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set Ω with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that Ω is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer problem to an arbitrary Riemannian manifold and also the posibility of replacing the condition on the domain to be a ball by [a] more general condition: to have a homogeneous boundary (i.e., boundary admitting a transitive group of isometries). We prove that if Ω has a homogeneous boundary, then (N) and (D) always admit solutions (in fact, for infinitely 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 Schiffer Conjecture or even its generalization.These ideas can be extended to other (essentially more complicated) overdetermined boundary value problems, including higher order equations and non-linear equations, which, in a number of important cases, may also have solutions in domains with isoparametric (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.
LA - eng
KW - Hipersuperficies; Variedad riemanniana; Variedad simétrica; Problema de contorno; Schiffer problem; isoparametric hypersurfaces; homogeneous boundary value problem; overdetermined boundary value problem
UR - http://eudml.org/doc/39616
ER -