2002 Fiscal Year Final Research Report Summary
Continuation and uniqueness for solutions of partial differential equations
Project/Area Number 
13640166

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Nagoya University 
Principal Investigator 
SUZUKI Noriaki Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50154563)

CoInvestigator(Kenkyūbuntansha) 
ISHIGE Kazuhiro Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90272020)
MIYAKE Masatake Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70019496)

Project Period (FY) 
2001 – 2002

Keywords  harmonic function / heat equation / Dirichlet problem / mean value theorem / heat ball 
Research Abstract 
We study the continuation and uniqueness for solutions of partial differential equations, by using potential theory. We have the following results. 1. In 2001, we showed a characterization of heat balls by mean value property for temperatures in Proc.Amer.Math.Soc. Then a generalization of it was obtained and published in Suriken Kokyuroku. Based on these results, we start to study the existence of mean value density for temperatures. In particular, a relation with the Dirichlet regularity and the existence of a bounded density or a density with positive infimum are discussed. The development of them is our new object of study. 2. We discueesd an extension of harmonic function on a domain. In the 2 dimensional case our problem is completely solved, but in higher dimensional case there are some problems. 3. We study a polynomial solution of the Dirichlet problem on a domain for the heat equation. In case that a domain is determined by a polynomial with degree less than 3, we obtain a necessary and sufficient condition under which the above problem is solvable. This result is publised in Bull.Aichi Inst.Tech. In case the degree is more than 4 we find that this problem is closely related with an assertion concerning to the zero points of Hermite polynomials. 4. In the connection of uniqueness of a solution of α parabolic operators on a half space, we study the Huygens property and the duality of parabolic Bergman spaces. Our spaces contain the usual harmonic and heat Bergman spaces.

Research Products
(8 results)