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Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems
 

Summary: Steklov Eigenproblems and the Representation of Solutions of
Elliptic Boundary Value Problems
Giles Auchmuty*
Department of Mathematics, University of Houston, Houston, Texas, USA
ABSTRACT
This paper describes some properties and applications of Steklov eigenproblems
for prototypical second-order elliptic operators on bounded regions in Rn
.
Results are described for Schroedinger and weighted harmonic equations. A vari-
ational description of the least eigenvalue leads to optimal L2
-trace inequalities. It
is shown that the eigenfunctions provide complete orthonormal bases of certain
closed subspaces of H1
O and also of L2
@O; ds. This allows the description,
and representation, of solution operators for homogeneous elliptic equations
subject to inhomogeneous Dirichlet, Neumann or Robin boundary data. They
are also used to describe Robin to Dirichlet and Neumann to Dirichlet operators
for these equations, and to describe the spectrum of these operators. The allow-
able regions are quite general; in particular classes of bounded regions with a

  

Source: Auchmuty, Giles - Department of Mathematics, University of Houston

 

Collections: Mathematics