Differential and Integral Equations

Decay transference and Fredholmness of differential operators in weighted Sobolev spaces

Patrick J. Rabier

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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.

Article information

Source
Differential Integral Equations, Volume 21, Number 11-12 (2008), 1001-1018.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1355502291

Mathematical Reviews number (MathSciNet)
MR2482494

Zentralblatt MATH identifier
1224.35137

Subjects
Primary: 35P05: General topics in linear spectral theory
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Citation

Rabier, Patrick J. Decay transference and Fredholmness of differential operators in weighted Sobolev spaces. Differential Integral Equations 21 (2008), no. 11-12, 1001--1018. https://projecteuclid.org/euclid.die/1355502291


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