Project/Area Number 
12440038

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KYOTO UN IVERSITY 
Principal Investigator 
MORIMOTO Yoshinori Kyoto Univ., Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (30115646)

CoInvestigator(Kenkyūbuntansha) 
TARAMA Shigeo Osaka City Univ., Faculty of Technology, Professor, 工学部, 教授 (90115882)
UEKI Naomasa Kyoto Univ., Graduate School of Human and Environmental Studies, Ass. Professor, 大学院・人間・環境学研究科, 助教授 (80211069)
TAKASAKI Kanehisa Kyoto Univ., Graduate School of Human and Environmental' Studies, Professor, 大学院・人間・環境学研究科, 教授 (40171433)
ANDO Hiroshi Ibaraki Univ., Faculty of Science, Assistant, 理学部, 助手 (60292471)
MORIOKA Tatsushi Osaka Educational Univ., Faculty of Education, Ass. Professor, 教育学部, 助教授 (80239631)
畑 政義 京都大学, 総合人間学部, 助教授 (40156336)
浅倉 史興 大阪電気通信大学, 工学部, 教授 (20140238)

Project Period (FY) 
2000 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥8,700,000 (Direct Cost: ¥8,700,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2002: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2001: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2000: ¥2,500,000 (Direct Cost: ¥2,500,000)

Keywords  infinitely degenerate / microlocal analysis / Sobolev inequality of logarithmic type / semiliner Dirichlet problem / FeffermanPhong inegality / Schrodinger equation / hypaellipticity / Wick calculus / 退化MongeAmpere方程式 / 対数型ソボレフ不等式 / 最大値原理 / Schodinger方程式 / key wards / Wick作用素 / 楕円型方程式 / 双極性 / 対数型 / sololev不等式 / 半線型楕円型方程式 / schodinger方程式 / 強一意性定理 / 双曲型方程式 
Research Abstract 
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 pseudodifferential 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 subelliptic. 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 FeffermanPhong 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 FeffermanPhong 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 densityofstates 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 ∂bLaplacian. 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
