arXiv 1907.080322019-07math.AP

非局部$p$ -Rayleigh商的渐近行为

Asymptotic behavior of nonlocal $p$-Rayleigh quotients

Feng Li

摘要：

让$N\geq 1$，$s,k\in(0,1)$，$p\in(1,\infty)$。设$t>1$，开有界集$Ω\subset\mathbb R^N$，$R$为$Ω$的半径.设$B_{tR}(Ω)$为包含$Ω$的球，其半径为$tR$，并且与$Ω$具有相同的中心。本文研究了第一$(s,p)$ -本征值和相应的第一$(s,p)$ -本征函数在逼近$k\rightarrow s$时的渐近性态。我们证明了在$k\rightarrow s^-$的不连续性和$k\rightarrow s^+$的连续性两个方向之间存在一种不同的现象，这种现象可以由具有正Besov容度的边界点上的本征函数的行为所触发.这种差异促使我们研究算子$(-Δ_p)^s$在不规则边界点上的边界行为。我们还刻画了当$k\rightarrow s^-$ .最后，基于P. Lindqvist和O.马迪奥证明通过将$\widetilde W^{s,p}_0(Ω)$约简为这里引入的所谓的相对非局部空间$\widetilde W^{s,p}_{0,tR}(Ω)$来进行，其等价于$\widetilde W^{s,p}_0(Ω)$，其中$\widetilde W^{s,p}_0(Ω)$被定义为$C^\infty_0(Ω)$在Gagliardo半范数$W^{s,p}(\mathbb R^N)$下的完备化，而$\widetilde W^{s,p}_{0,tR}(Ω)$被定义为$C^\infty_0(Ω)$在Gagliardo半范数$W^{s,p}(B_{tR}(Ω))$下的完备化。作为部分结果，我们建立了算子$(-Δ_p)^s$在$\widetilde W^{s,p}_0(Ω)$及其对偶空间$\widetilde W^{-s,p^\prime}(Ω)$之间的同胚，其中$1/p+1/p^\prime=1$ .

英文摘要：

Let $N\geq 1$, $s,k\in(0,1)$, $p\in(1,\infty)$. Let $t>1$, open bounded set $Ω\subset\mathbb R^N$, $R$ be the radius of $Ω$. Let $B_{tR}(Ω)$ be the ball containing $Ω$ with radius $tR$ and with the same center as $Ω$. In this article we study the asymptotic behavior of the first $(s,p)$-eigenvalue and corresponding first $(s,p)$-eigenfunctions during the approximation $k\rightarrow s$. We show that there exhibits a different phenomenon between the two directions of discontinuity of $k\rightarrow s^-$ and continuity of $k\rightarrow s^+$, which can be triggered by behaviors of eigenfunctions on the boundary points bearing the positive Besov Capacity. And this difference prompts us to study the boundary behavior of operators $(-Δ_p)^s$ on the irregular boundary points. We also characterize some equivalent forms of the continuity case when $k\rightarrow s^-$. In the end, we construct a counterexample for the discontinuity case during $k\rightarrow s^-$ based on the positivity of Besov capacity of Cantor set and the fine decay estimates up to the regular boundary points, used by P. Lindqvist and O. Martio. The proof works by reducing $\widetilde W^{s,p}_0(Ω)$ to the so-called Relative-nonlocal spaces $\widetilde W^{s,p}_{0,tR}(Ω)$ introduced here, which is equivalent to $\widetilde W^{s,p}_0(Ω)$, where $\widetilde W^{s,p}_0(Ω)$ is defined as the completion of $C^\infty_0(Ω)$ under the Gagliardo semi-norm $W^{s,p}(\mathbb R^N)$, and $\widetilde W^{s,p}_{0,tR}(Ω)$ defined as the completion of $C^\infty_0(Ω)$ under the Gagliardo semi-norm $W^{s,p}(B_{tR}(Ω))$. As a partial result, we established the Homemorphism of the operator $(-Δ_p)^s$ between $\widetilde W^{s,p}_0(Ω)$ and its dual space $\widetilde W^{-s,p^\prime}(Ω)$, where $1/p+1/p^\prime=1$.

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