2008 Decay transference and Fredholmness of differential operators in weighted Sobolev spaces
Patrick J. Rabier
Differential Integral Equations 21(11-12): 1001-1018 (2008). DOI: 10.57262/die/1355502291


We show that, for some family of weights $\omega ,$ there are corresponding weighted Sobolev spaces $W_{\omega }^{m,p}$ on $ \mathbb {R}^{N}$ such that whenever $P(x,\partial)$ is a differential operator with $L^{\infty }$ coefficients and $P(x,\partial):W^{m,p}\rightarrow L^{p}$ is Fredholm for some $p\in (1,\infty),$ then $P(x,\partial):W_{\omega }^{m,p}\rightarrow L_{\omega }^{p}$ ($=W_{\omega }^{0,p}$) remains Fredholm with the same index. We also show that many spectral properties of $P(x,\partial)$ are closely related, or even the same, in the non-weighted and the weighted settings. The weights $\omega $ arise naturally from a feature of independent interest of the Fredholm differential operators in classical Sobolev spaces (``full'' decay transference), proved in the preparatory Section 2. A main virtue of the spaces $W_{\omega }^{m,p}$ is that they are well suited to handle nonlinearities that may be ill-defined or ill-behaved in non-weighted spaces. Together with the invariance results of this paper, this has proved to be instrumental in resolving various bifurcation issues in nonlinear elliptic PDEs.


Download Citation

Patrick J. Rabier. "Decay transference and Fredholmness of differential operators in weighted Sobolev spaces." Differential Integral Equations 21 (11-12) 1001 - 1018, 2008. https://doi.org/10.57262/die/1355502291


Published: 2008
First available in Project Euclid: 14 December 2012

zbMATH: 1224.35137
MathSciNet: MR2482494
Digital Object Identifier: 10.57262/die/1355502291

Primary: 35P05
Secondary: 46E35 , 47A53 , 47F05

Rights: Copyright © 2008 Khayyam Publishing, Inc.


This article is only available to subscribers.
It is not available for individual sale.

Vol.21 • No. 11-12 • 2008
Back to Top