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Decay transference and Fredholmness of differential operators in weighted Sobolev spaces
Differential and Integral Equations ( IF 1.4 ) Pub Date : 2012-12-14
Patrick J. Rabier

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.

中文翻译:

加权Sobolev空间中微分算子的衰减转移和Fredholmness。

我们表明,对于某些权重$ \ omega,$在$ \ mathbb {R} ^ {N} $上有相应的加权Sobolev空间$ W _ {\ omega} ^ {m,p} $ (x,\ partial)$是具有$ L ^ {\ infty} $系数和$ P(x,\ partial):W ^ {m,p} \ rightarrow L ^ {p} $的微分算子$ p \ in(1,\ infty),$然后$ P(x,\ partial):W _ {\ omega} ^ {m,p} \ rightarrow L _ {\ omega} ^ {p} $($ = W_ { \ omega} ^ {0,p} $)仍然是Fredholm的相同索引。我们还表明,在非加权和加权设置中,$ P(x,\ partial)$的许多光谱特性密切相关,甚至相同。权重$ \ omega $自然来自于经典Sobolev空间中Fredholm微分算子的独立兴趣(“完全”衰减传递),这在准备部分2中得到了证明。空间$ W _ {\欧米茄} ^ {m,p} $是它们非常适合处理在非加权空间中可能定义不正确或行为不正确的非线性。连同本文的不变性结果一起,这已被证明有助于解决非线性椭圆PDE中的各种分叉问题。
更新日期:2012-12-14
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