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

基本信息

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

项目摘要

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

本研究的目的是利用伪微分算子、傅里叶积分算子、调和分析和随机微积分的理论，研究简并椭圆算子的正性如何体现在偏微分方程的解的结构中。与国外联合研究人员徐教授共同研究了某些二阶主部分无限退化的半线性椭圆方程的Dirichlet问题。给出了该问题的解，其次阐明了解的连续性和 C 正则性，与有限简并椭圆算子的情况不同，只能对某些具有弱正性的无限简并椭圆算子进行对数正则性向上估计。在这种对数正则性向上估计的假设下，我们推导了对数型Sobolev不等式，并通过求解证明了狄利克雷问题解的存在性相关的变分问题的解的有界性、连续性和 C∞ 正则性的证明与用于主要部分是椭圆或次椭圆的半线性方程的传统方法完全不同。对于线性无限简并椭圆算子的 C∞-亚椭圆性，J.-M.Bony 和 J.-M.Bony 的最新结果与简并椭圆算子的正性有关。对D.Tataru进行了检验，讨论了关于伪微分算子正性的Fefferman-Phong不等式，与首次引入Wick微积分研究主型伪微分算子可解性的Lerner教授进行了联合研究，结果表明： Wick 演算也适用于 Fefferman-Phong 不等式的证明，而不是 Tataru 论文中使用的 FBI 算子。我们的另一个证明是在与联合工作中改进的 Wick 算子的乘积公式中进行的。安藤研究员研究了与无限简并微局域分析相关的随机磁场薛定谔算子的谱，发现泡利哈密顿量的状态密度函数与前一种情况具有异常不同的结构。，并将这些结果应用到∂b-拉普拉斯的亚椭圆性研究中，研究者从偏微分方程的微局部分析的角度考虑了二阶方程的Goursat问题。 Tarama 通过能量估计扩展了长谷川的结果，研究人员 Takasaki 研究了孤子方程特定解的代数几何结构，与简并椭圆方程的奇异解相关。