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2006-06

Regions of positivity for polyharmonic Green functions in arbitrary domains

by Grunau, Hans-Christoph, Sweers, Guido.


Series: 2006-06, Preprints

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles
35J40 Boundary value problems for higher-order elliptic equations

Abstract:
The Green function for the biharmonic operator on bounded domains
with zero Dirichlet boundary conditions is in general not of fixed
sign. However, by extending an idea of Z.~Nehari, we are able to
identify regions of positivity for Green functions of polyharmonic
operators. In particular, the biharmonic Green function is
considered in all space dimensions. As a consequence we see that the
negative part of any such Green function is somehow small compared
with the singular positive part.

Keywords:
polyharmonic Green function; regions of positivity; general domains

This paper was published in:
Proc. Amer. Math. Soc. 135, 3537-3546 (2007).