论文标题
在装有功率重量的半线上定义的加权Sobolev空间中的密度结果和微量操作员
Density results and trace operator in weighted Sobolev Spaces defined on the half line equipped with power weights
论文作者
论文摘要
我们研究$ w_0^{1，p}（\ Mathbb {r} _+，t^β）$的属性 - 在Power -C_0^\ Infty（\ Mathbb {r} _+）$ power -weighted sobolev sobolev sobolev spaces $ w^^{1，p}（$ mathb}（$ mathb}）中， $β\ in \ Mathbb {r} $。除其他结果外，我们获得了$ W_0^{1，p}的分析表征（\ Mathbb {r} _+，t^β）$ in \ in \ mathbb {r} $。我们的分析基于精确的研究两个跟踪操作员：$ tr^{0}（u）：= \ lim_ {t \ to 0} u（t）$和$ tr^{\ infty}（u）：= \ lim_ {t \ t \ to \ to \ fto \ infty} u（t） $ w_0^{1，p}（\ mathbb {r} _+，t^β）$接近零或无穷大。获得的陈述可以有助于与径向对称性的ODE或PDE中的边界值问题正确提出。我们还可以将结果应用于复杂插值理论的某些问题，该理论由M. Cwikel和A. Einav在2019年提出，我们在Sobolev Space $ W^{1，P}的特定情况下进行了讨论（\ Mathbb {r} _+，T^β）$。
We study properties of $W_0^{1,p}(\mathbb{R}_+,t^β)$ - the completion of $C_0^\infty(\mathbb{R}_+)$ in the power-weighted Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^β)$, where $β\in\mathbb{R}$. Among other results, we obtain the analytic characterization of $W_0^{1,p}(\mathbb{R}_+,t^β)$ for all $β\in \mathbb{R}$. Our analysis is based on the precise study the two trace operators: $Tr^{0}(u):= \lim_{t\to 0} u(t)$ and $Tr^{\infty}(u):= \lim_{t\to \infty} u(t)$, which leads to the analysis of the asymptotic behavior of functions from $W_0^{1,p}(\mathbb{R}_+,t^β)$ near zero or infinity. The obtained statements can contribute to the proper formulation of Boundary Value Problems in ODE's, or PDE's with the radial symmetries. We can also apply our results to some questions in the complex interpolation theory, raised by M. Cwikel and A. Einav in 2019, which we discuss within the particular case of Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^β)$.
