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

这项研究的目的是通过使用伪差异操作员，傅立叶积分运算符，谐波分析和随机计算的理论来研究偏椭圆算子的阳性如何反映到偏微分方程的解决方案的结构。首席研究员认为，通过与外国人联合研究人员的Xu教授的联合研究，二阶的主要部分无限地退化，其主要部分无限地退化。首先，显示了解决该问题的解决方案的存在和界限，其次，证明了解决方案的连续性和C∞规则性。对数规律性提高估计只有对于某些无限退化的椭圆运算符，其积极性较弱，这与有限变性的椭圆运算符的案例有所不同。在这种对数规律性提高估计值的假设下，我们得出了Logari的Sobolev不平等……更多的Thmic类型，并通过解决相关的变异问题来证明存在Dirichlet问题的解决方案。对我们问题的界限，连续性和C∞规律性的证明与用于半平衡的传统方法完全不同，该方法的主要部分是椭圆形或亚椭圆形的。我们的方法基于线性无限退化椭圆运算符的C∞-纤维纤维素的技术。关于退化的椭圆运算符的潜力，研究了J.-M.Bony和d.tataru的最新结果，其中讨论了Fefferman-Phong的不平等现象，讨论了有关假数分支机构的积极性的不平等。作为与Lerner教授的共同研究，他首先引入了Wick Cyculus，以研究主要类型的假差异操作员的可溶性，我们表明，Wick演算也适用于Fefferman-Phong-Phong不平等现象，而不是塔塔鲁（Tataru）论文中雇用的FBI运营商的证明。我们的另一个证明是在与安多（Ando）联合合作中获得的Wick运算符的产品配方进行改进。一个研究者Ueki研究了与微局部分析有关的随机磁场的频谱，并无限退化，发现在前病例中Pauli Hamiltonian的结构中，态密度的功能非常不同，并将这些结果应用于低纤维化性的结果，以研究B-LaplacIan。从针对部分微分方程的微局部分析的角度来看，研究人员塔拉玛（Tarama）考虑了二阶方程式的二阶方程问题，塔拉玛（Tarama）考虑了通过能量估计来扩展谷伐的结果，并且在研究人员Takanele solation to nequalite for necorelipt方程中研究了索利顿方程的代数几何结构对索利顿方程的特定解决方案。较少的