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Existence of the gauge for fractional Laplacian Schr\"{o}dinger operators
Analysis of PDEs · Submitted 20 Feb 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<\alpha <1$.
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The abstract, sentence by sentence
Let $\Omega \subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $\omega $ is a locally finite Borel measure on $\Omega$. For $\alpha \in (0,2)$, define the fractional Laplacian $(-\triangle )^{\alpha/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $\alpha$ on $\Omega$. Define $T(u) = G(u \omega).$ If $\Vert T \Vert_{L^2(\omega) \rightarrow L^2 (\omega)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{\alpha/2, 2}_0 (\Omega)$ of \[ (-\triangle)^{\alpha/2} u = u \omega + \nu \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, \Omega^c, \] for $\nu$ in the dual Sobolev space $L^{-\alpha/2, 2} (\Omega)$. If $\Omega$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $\nu = \chi_{\Omega}$. These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 \omega)$ corresponding to the problem \[ (-\triangle)^{\alpha/2} u = u \omega \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, \Omega^c . \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<\alpha <1$. For $1 \leq \alpha <2$, 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 $1 < \alpha <2$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\Omega \subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. | background 0.99 | 0.08 | 0.01 | asserted 0.84 |
| s1 | Suppose $\omega $ is a locally finite Borel measure on $\Omega$. | background 0.93 | 0.11 | 0.01 | asserted 0.87 |
| s2 | For $\alpha \in (0,2)$, define the fractional Laplacian $(-\triangle )^{\alpha/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $\alpha$ on $\Omega$. | background 0.60 | 0.22 | 0.02 | asserted 0.90 |
| s3 | Define $T(u) = G(u \omega).$ If $\Vert T \Vert_{L^2(\omega) \rightarrow L^2 (\omega)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{\alpha/2, 2}_0 (\Omega)$ of \[ (-\triangle)^{\alpha/2} u = u \omega + \nu \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, \Omega^c, \] for $\nu$ in the dual Sobolev space $L^{-\alpha/2, 2} (\Omega)$. | result 0.76 | 0.81 | 0.12 | asserted 1.00 |
| s4 | If $\Omega$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $\nu = \chi_{\Omega}$. | result 0.91 | 0.86 | 0.08 | asserted 1.00 |
| s5 | These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 \omega)$ corresponding to the problem \[ (-\triangle)^{\alpha/2} u = u \omega \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, \Omega^c . | contribution 0.81 | 0.72 | 0.14 | asserted 0.91 |
| s6 | \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<\alpha <1$. | result 0.99 | 0.93 | 0.30 | asserted 1.00 |
| s7 | For $1 \leq \alpha <2$, a solution exists if the norm of $T$ is sufficiently small. | result 0.99 | 0.92 | 0.19 | asserted 0.95 |
| s8 | We also show that the condition $\Vert T \Vert <1$ does not imply the existence of a solution if $1 < \alpha <2$. | result 0.97 | 0.93 | 0.23 | asserted 1.00 |
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Original abstract
Let $\Omega \subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $\omega $ is a locally finite Borel measure on $\Omega$. For $\alpha \in (0,2)$, define the fractional Laplacian $(-\triangle )^{\alpha/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $\alpha$ on $\Omega$. Define $T(u) = G(u \omega).$ If $\Vert T \Vert_{L^2(\omega) \rightarrow L^2 (\omega)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{\alpha/2, 2}_0 (\Omega)$ of \[ (-\triangle)^{\alpha/2} u = u \omega + \nu \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, \Omega^c, \] for $\nu$ in the dual Sobolev space $L^{-\alpha/2, 2} (\Omega)$. If $\Omega$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $\nu = \chi_{\Omega}$. These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 \omega)$ corresponding to the problem \[ (-\triangle)^{\alpha/2} u = u \omega \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, \Omega, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, \Omega^c . \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<\alpha <1$. For $1 \leq \alpha <2$, 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 $1 < \alpha <2$.
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2 arXiv papers cite this in 3 sentences
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Cites it as background 1 paper
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For the fractional Schrödinger operator $(-)^α/2 -ω$, criteria of the existence of the gauge $u_1$ in the case $0<α<2$ were obtained in [ref] .
Positive solutions and harmonic measure for Schr\"{o}dinger operators in uniform domains · Introduction · 0.84
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