区间上分数阶拉普拉斯算子的内伯努利自由边界问题

IF 0.7 2区 数学 Q2 MATHEMATICS
Tadeusz Kulczycki, Jacek Wszoła
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In particular, we show that there exists a constant $$\\lambda _{\\alpha ,D} &gt; 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (called the Bernoulli constant) such that the problem has no solution for $$\\lambda \\in (0,\\lambda _{\\alpha ,D})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , at least one solution for $$\\lambda = \\lambda _{\\alpha ,D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and at least two solutions for $$\\lambda &gt; \\lambda _{\\alpha ,D}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . 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引用次数: 0

摘要

研究了区间D上参数为$$\lambda > 0$$ λ &gt的$$(-\Delta )^{\alpha /2}$$ (- Δ) α / 2的内Bernoulli自由边界问题解的结构;0。特别地,我们证明了存在一个常数$$\lambda _{\alpha ,D} > 0$$ λ α, D &gt;0(称为伯努利常数)使得问题对于$$\lambda \in (0,\lambda _{\alpha ,D})$$ λ∈(0,λ α, D)无解,对于$$\lambda = \lambda _{\alpha ,D}$$ λ = λ α, D至少有一个解,对于$$\lambda > \lambda _{\alpha ,D}$$ λ &gt至少有两个解;λ α, d。我们还研究了具有一个自由边界点的区间分数阶拉普拉斯函数的内伯努利问题。讨论了伯努利问题与相应的变分问题的联系,并提出了一些猜想。特别地,当$$\alpha = 1$$ α = 1时,我们证明了$$(-\Delta )^{\alpha /2}$$ (- Δ) α / 2的内伯努利自由边界问题在一个区间上的解不是相应变分问题的极小解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval

On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval
Abstract We study the structure of solutions of the interior Bernoulli free boundary problem for $$(-\Delta )^{\alpha /2}$$ ( - Δ ) α / 2 on an interval D with parameter $$\lambda > 0$$ λ > 0 . In particular, we show that there exists a constant $$\lambda _{\alpha ,D} > 0$$ λ α , D > 0 (called the Bernoulli constant) such that the problem has no solution for $$\lambda \in (0,\lambda _{\alpha ,D})$$ λ ( 0 , λ α , D ) , at least one solution for $$\lambda = \lambda _{\alpha ,D}$$ λ = λ α , D and at least two solutions for $$\lambda > \lambda _{\alpha ,D}$$ λ > λ α , D . We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for $$\alpha = 1$$ α = 1 that there exists solutions of the interior Bernoulli free boundary problem for $$(-\Delta )^{\alpha /2}$$ ( - Δ ) α / 2 on an interval which are not minimizers of the corresponding variational problem.
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来源期刊
Collectanea Mathematica
Collectanea Mathematica 数学-数学
CiteScore
2.70
自引率
9.10%
发文量
36
审稿时长
>12 weeks
期刊介绍: Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.
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