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Regions of positivity for polyharmonic Green functions in arbitrary domains

by Grunau, Hans-Christoph, Sweers, Guido.

Series: 2006-06, Preprints

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

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.

polyharmonic Green function; regions of positivity; general domains

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