Continuation and uniqueness for solutions of partial differential equations

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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.