the positivity of degenerate elliptic operators and the microlocal analysis on solutions for partial differentiai equations

简并椭圆算子的正性及偏微分方程解的微局域分析

• 批准号: 12440038
• 负责人: MORIMOTO Yoshinori
• 金额: $ 5.57万
• 依托单位: KYOTO UN IVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440038/
• 关键词: infinitely degenerate; microlocal analysis; Sobolev inequality of logarithmic type; semiliner Dirichlet problem; Fefferman-Phong inegality; Schrodinger equation; hypaellipticity; Wick calculus; 退化Monge-Ampere方程式; 対数型ソボレフ不等式; 最大値原理; Schodinger方程

The purpose of this research is to study how the positivity of degenerate elliptic operators is reflected to the structure of solutions for partial differential equations, by using the theories of pseudo-differential operators, Fourier integral operators, harmonic analysis and stochastic calculus. Head investigator considered the Dirichlet problem for certain semilinear elliptic equations whose principal parts of second order degenerate infinitely, by joint research with Prof. Xu who is a foreigner joint research person. Firstly, the existence and the boundedness of solutions to this problem were shown, and secondly the continuity and C∞ regularity of solutions were clarified. The logarithmic regularity up estimate can be only expected for certain infinitely degenerate elliptic operators with weak positivity, differing from the case for elliptic operators with finite degeneracy. Under the assumption of this logarithmic regularity up estimate, we derived the Sobolev inequality of logari … More thmic type, and proved the existence of solutions to the Dirichlet problem by solving the associated variational problem. The proofs of the boundedness, the continuity and C∞ regularity of solutions to our problem are completely different from the traditional methods used for semilinear equations whose principal part is elliptic or sub-elliptic. Our method is based on the technique for C∞-hypoellipticity for linear infinitely degenerate elliptic operators. In relation to the positivity of degenerate elliptic operators, the recent results of J.-M.Bony and D.Tataru were examined, where the inequality of Fefferman-Phong concerning the positivity of pseudodifferential operators are discussed. As a joint research with Prof. Lerner who introduced firstly Wick calculus for the research of solvability of pseudodifferential operators of principal type, we showed that the Wick calculus is also applicable to the proof of Fefferman-Phong inequality instead of FBI operators employed in Tataru's paper. Our another proof is carried out in refining the product formula of Wick operators obtained in the joint work with Ando. An investigator Ueki studied the spectrum of a Schrodinger operator with the random magnetic field relevant to the microlocal analysis with infinitely degeneracy, found out that a density-of-states function have remarkably different structure in the case of Pauli Hamiltonian from the former case, and applied those results to research of the hypoellipticity for ∂b-Laplacian. From the point of view on the microlocal analysis for partial differential equations, the Goursat problem to the second order equation was considered by an investigator Tarama who extended Hasegawa's result by energy estimates, and the algebraic geometry structure of the particular solution to soliton equations was studied by an investigator Takasaki, in relation to the singular solutions for degenerate elliptic equations. Less

利用拟微分算子、Fourier积分算子、调和分析和随机微积分的理论，研究退化椭圆算子的正性如何反映到偏微分方程解的结构中.本课题组组长与外籍合作研究人员徐教授共同研究了一类二阶主部无限退化的半线性椭圆型方程的Dirichlet问题。首先证明了该问题解的存在性和有界性，其次阐明了解的连续性和C∞正则性.对数正则性上估计仅对某些弱正性的无限退化椭圆型算子才有，这与有限退化椭圆型算子的情形不同。在这种对数正则性上估计的假设下，我们得到了logari的Sobolev不等式 ...更多信息 thmic型，并通过求解相关的变分问题证明了狄利克雷问题解的存在性。我们的问题解的有界性、连续性和C∞正则性的证明与传统的主要部分为椭圆或次椭圆的半线性方程的证明方法完全不同。我们的方法是基于线性无限退化椭圆算子的C∞-亚椭圆性技巧。关于退化椭圆算子的正性，J.- M.Bony和D.Tataru的结果，其中讨论了关于伪微分算子正性的Fézumman-Phong不等式.作为与Lerner教授的合作研究，我们证明了Wick演算代替Tataru论文中使用的FBI算子同样适用于Festhiman-Phong不等式的证明。Lerner教授首次将Wick演算引入到主型伪微分算子的可解性研究中。我们的另一个证明是在改进与Ando合作得到的Wick算子的乘积公式时进行的。研究者植木研究了薛定谔算符在与无限简并的微局域分析相关的随机磁场作用下的谱，发现在泡利哈密顿量的情况下，态密度函数的结构与前一种情况有显著不同，并将这些结果应用于研究泡利-拉普拉斯算子的亚椭圆性。从偏微分方程的微局部分析的观点出发，Tarama研究了二阶方程的Goursat问题，他通过能量估计推广了长谷川的结果，高崎研究了孤子方程特解的代数几何结构，并与退化椭圆型方程的奇异解有关.少

项目成果

森本芳則: "Remark on the analytic smoothing for the Schrodinger equation"Indiana Univ.Math.. (未定).

Yoshinori Morimoto：“关于薛定谔方程的解析平滑的评论”印第安纳大学数学..（待定）。

多羅間茂雄: "On the estimate of some conjugation"Mem.Fac.Eng.Osaka City Univ.. 41巻. 117-123 (2000)

Shigeo Tarama：“关于某些共轭的估计”Mem.Fac.Eng.Osaka City Univ.. 41. 117-123 (2000)

Yoshinori Morimoto, Chao-Jiang Xu: "Regularity of weak solution for a class of infinitely degenerate ellitpic semilinear equations,"Seminaire Equations aux Derivees Partielles Ecole Polytechnique. VII-1-VII-14 (2003)

Yoshinori Morimoto、Chao-Jiang Xu：“一类无限退化椭圆半线性方程的弱解的正则性”，高等理工学院派生方程研讨会。

畑 政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric, function"Analytic Number Theory, Developments in Mathematics. 6巻. 157-172 (2002)

Masayoshi Hata：“高斯超几何函数的对数导数的帕德逼近”《解析数论》，《数学进展》第 6 卷。157-172 (2002)。

森本芳則, C-J.Xu: "Regularity of weak solution for a class of infinitely degenerate ellitpic semilinear equations"Seminaire Equations aux Derivees Partielles Ecole Polytechnique.2003-2004. VII-1-VII-14 (2004)

森本嘉德，C-J。