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

本文利用伪微分算子、傅立叶积分算子、调和分析和随机微积分等理论，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。首席研究员与外籍联合研究员徐教授共同研究了一类二阶主成分无限退化的半线性椭圆方程的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。与有限简并椭圆算子的情况不同，对数正则性上估计仅对某些弱正无穷简并椭圆算子才有期望。在这种对数正则性上估计的假设下，我们导出了logi…More thmic型的Sobolev不等式，并通过求解相关变分问题证明了Dirichlet问题解的存在性。该问题解的有界性、连续性和C∞正则性的证明，与传统的主部为椭圆型或次椭圆型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∞-亚椭圆性技术。关于简并椭圆算子的正性，本文给出了j - m。本文讨论了伪微分算子正性的Fefferman-Phong不等式。我们与首先引入Wick演算来研究主型伪微分算子的可解性的Lerner教授共同研究，证明了Wick演算也适用于证明Fefferman-Phong不等式，而不是Tataru论文中使用的FBI算子。我们的另一个证明是在改进与安藤联合工作中得到的Wick算子的乘积公式的过程中进行的。研究者Ueki研究了与微局部分析相关的随机磁场下薛定谔算子的谱，发现泡利哈密顿量下的态密度函数的结构与前一种情况有明显的不同，并将这些结果应用于∂b-拉普拉斯量的亚椭圆性研究。从偏微分方程的微局部分析的角度出发，Tarama研究了二阶方程的Goursat问题，并用能量估计推广了Hasegawa的结果，Takasaki研究了孤子方程的特解的代数几何结构与退化椭圆方程的奇异解的关系。少

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