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arXiv 1802.07173math.APmath.CA
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分数阶Laplacian Schrodinger算子规范的存在性

Existence of the gauge for fractional Laplacian Schrödinger operators

Michael W. Frazier, Igor E. Verbitsky

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31 pages
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摘要：

设$Ω\subseteq \mathbb{R}^n$为开集，其中$n \geq 2$ .假设$ω$是$Ω$上的局部有限Borel测度。对于$α\in (0,2)$，通过$\mathbb{R}^n$上的傅里叶变换定义分数拉普拉斯算子$(-\triangle )^{α/2}$，并令$G $为$Ω$上相应的$α$阶格林算子。定义$T(u) = G(u ω).$如果$\Vert T \Vert_{L^2(ω) \rightarrow L^2 (ω)} <1$，我们得到了在对偶Sobolev空间$L^{-α/2, 2} (Ω)$中的$ν$的\[ (-\triangle)^{α/2} u = u ω+ ν\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, Ω^c, \]的齐次Sobolev空间$L^{α/2, 2}_0 (Ω)$中的唯一弱解$u$的表示.如果$Ω$是一个有界的$C^{1,1}$域，这种表示会产生匹配的指数上下逐点估计的解决方案时，$ν= χ_Ω$。这些估计被用来研究的存在性的解决方案$u_1$（称为“规范”）的积分方程$u_1=1+G(u_1 ω)$对应的问题\[ (-\triangle)^{α/2} u = u ω\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, Ω^c . \]我们表明，如果$\Vert T \Vert <1$，那么$u_1$总是存在，如果$0<α<1$。对于$1 \leq α<2$，如果$T$的范数足够小，则存在解。我们还表明，条件$\Vert T \Vert <1$并不意味着一个解决方案的存在，如果$1 < α<2$。

英文摘要：

Let $Ω\subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $ω$ is a locally finite Borel measure on $Ω$. For $α\in (0,2)$, define the fractional Laplacian $(-\triangle )^{α/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $α$ on $Ω$. Define $T(u) = G(u ω).$ If $\Vert T \Vert_{L^2(ω) \rightarrow L^2 (ω)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{α/2, 2}_0 (Ω)$ of \[ (-\triangle)^{α/2} u = u ω+ ν\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, Ω^c, \] for $ν$ in the dual Sobolev space $L^{-α/2, 2} (Ω)$. If $Ω$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $ν= χ_Ω$. These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 ω)$ corresponding to the problem \[ (-\triangle)^{α/2} u = u ω\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, Ω^c . \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<α<1$. For $1 \leq α<2$, a solution exists if the norm of $T$ is sufficiently small. We also show that the condition $\Vert T \Vert <1$ does not imply the existence of a solution if $1 < α<2$.

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