分数阶Helmholtz方程的复解和实值解

IF 1.8 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-07-04 DOI:10.1002/mma.11197

Zifei Shen, Shuijin Zhang

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引用次数: 0

摘要

本文讨论分数阶亥姆霍兹方程的极限吸收原理：（−Δ） s u−λ u = f (x，U)，在n $$ {\left(-\Delta \right)}&amp;#x0005E;su-\lambda u&amp;#x0003D;f\left(x,u\right),\kern0.60em \mathrm{in}\kern0.60em {\mathbb{R}}&amp;#x0005E;n $$中，其中n≥3，0 ＆lt; λ ＆lt; +∞$$ n\ge 3,\kern0.3em 0&lt;\lambda &lt;&amp;#x0002B;\infty $$和n n + 1 ＆lt； s＆lt; n2 $$ \frac{n}{n&amp;#x0002B;1}&lt;s&lt;\frac{n}{2} $$是两个实参数。通过建立分数阶Helmholtz算子解的有界性估计，我们得到了非平凡的lq (n) $$ {L}&amp;#x0005E;q\left({\mathbb{R}}&amp;#x0005E;n\right) $$复形分数阶亥姆霍兹方程的值解。通过建立对偶变分框架，利用非消失原理得到分数阶亥姆霍兹方程的实值解。

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Complex and Real Valued Solutions for Fractional Helmholtz Equation

In this paper, we are concerned with the limiting absorption principle for the fractional Helmholtz equation: ${(- Δ)}^{s} u - λ u = f (x, u), in ℝ^{n}$ , where $n \geq 3, 0 < λ < + \infty$ and $\frac{n}{n + 1} < s < \frac{n}{2}$ are two real parameters. By establishing the boundedness estimate for the resolvent of fractional Helmholtz operator, we obtain the nontrivial $L^{q} (ℝ^{n})$ complex valued solutions for the fractional Helmholtz equation. By setting up a dual variational framework, we also obtain the real valued solutions for the fractional Helmholtz equation via a non-vanishing principle.

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.