Continuation and uniqueness for solutions of partial differential equations

本文利用位势理论研究了偏微分方程解的连续性和唯一性。我们有以下结果。1. 2001年，我们在Proc.Amer.Math. Soc中用温度的平均值性质给出了热球的一个特征，然后得到了它的一个推广，并发表在Suriken Kokyuroku中。在此基础上，我们开始研究温度平均值密度的存在性。特别地，讨论了Dirichlet正则性与有界密度或具有正下确界的密度的存在性的关系。他们的发展是我们新的研究对象.讨论了整环上调和函数的一种扩张.在2维的情况下，我们的问题是完全解决的，但在高维的情况下，还有一些问题.研究了一类热传导方程在区域上的Dirichlet问题的多项式解.在整环由次数小于3的多项式确定的情况下，得到了上述问题可解的一个充要条件。该结果发表在Bull.Aichi Inst.Tech中。在次数大于4的情况下，我们发现这个问题与关于Hermite多项式零点的一个断言密切相关.本文结合半空间上α抛物算子解的唯一性，研究了抛物Bergman空间的Huygens性质和对偶性.我们的空间包含通常的调和和热伯格曼空间。

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.