Monotonicity of the inverse of weakly elliptic differential operators

Herzog, Gerd
Lemmert, Roland

facul_01; 14

Abstract:
We consider a weakly elliptic differential operator
$$Lu:= \sum_{j,k=1}^{n}a{j,k}D_{j}D_{k}u + \sum_{j=1}^{n}b_{j}D_{j}u + cu$$
in the Fr\'echet space $F$ of all $C^{\infty}$ functions
$u: \mathbb{R}^{n} \rightarrow \mathbb{R}$ with $u$ and all
derivatives of $u$ bounded. We prove in case $c < 0$ that $L:F \rightarrow F$
is invertible and $L^{-1}$ is monotone decreasing with respect
to each ordering on $F$ which is defined by a shift invariant
wedge. This result can be applied to obtain informations on the
behaviour as $\| x \| \rightarrow \infty$ of bounded entire solutions
of $Lu=v$ and related nonlinear problems.

Primary MSC: 35R45 Partial differential inequalities
Secondary MSC: 58G35 Invariance and symmetry properties