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

Abstract

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.

Citation

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Patrick J. Rabier. "Decay transference and Fredholmness of differential operators in weighted Sobolev spaces." Differential Integral Equations 21 (11-12) 1001 - 1018, 2008.

Information

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

zbMATH: 1224.35137
MathSciNet: MR2482494

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

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.21 • No. 11-12 • 2008
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