Asymptotic analysis of fundamental solutions of hypoelliptic operators

IF 0.8 4区数学 Q2 MATHEMATICS

Georgian Mathematical Journal Pub Date : 2023-10-28 DOI:10.1515/gmj-2023-2072

George Chkadua, Eugene Shargorodsky

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引用次数: 0

Abstract

Abstract Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 𝐏 ⁢ ( i ⁢ ∂ x ) = ( P 1 ⁢ ( i ⁢ ∂ x ) ) m 1 ⁢ ⋯ ⁢ ( P l ⁢ ( i ⁢ ∂ x ) ) m l \mathbf{P}(i\partial_{x})=(P_{1}(i\partial_{x}))^{m_{1}}\cdots(P_{l}(i\partial% _{x}))^{m_{l}} with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 𝐏 ⁢ ( i ⁢ ∂ x ) ⁢ u = f {\mathbf{P}(i\partial_{x})u=f} in ℝ n {\mathbb{R}^{n}} . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation.

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次椭圆算子基本解的渐近分析

摘要研究了具有实数零的特征多项式的准椭圆偏微分算子𝐏(i∂x)=(p1¹¹(i∂x)) m 1¹⋯⋯(p1¹¹(i∂x)) m l \mathbf{P}(i\partial_{x})=(P_{1}(i\partial_{x}))^{m_{1}}\cdots(P_{l}(i\partial% _{x}))^{m_{l}}的基本解在无穷远处的渐近行为。基于基本解的渐近展开式，给出了函数的渐近类，并建立了方程𝐏(i∑∂x)∑u=f {\mathbf{P}(i\partial_{x})u=f}在∈n {\mathbb{R}^{n}}中的解的存在唯一性。所得结果特别暗示了经典亥姆霍兹方程的一个新的唯一性定理。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Georgian Mathematical Journal 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.