Research of an integration representation of the solutions of the partial differential equation of elliptic type in unbounded domains and its stochastic analysis consideration

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Consideration of an integration representation of the solution of the Dirichlet problem and the Neumann problem with respect to the boundary value problem of an elliptic partial differential equation on unbounded domains was the purpose. About the corn and the cylinder, the result has already been obtained. About another unbounded domain strip, although the integration representation of a special solution has already been obtained also the concrete composition of a general solution and the problem of uniqueness of a certain kind are still unsolved. However, many re-search results of the action of the harmonic function which is a solution were able to be obtained. Namely, minimally thin sets are the potential theoretical exceptional sets which were studied in detail by Doob etc. Rarefied sets are the exceptional sets studied by Ahlfors, Hayman, etc. from the function theoretical viewpoint about the degree of increase of a function. It is usual that the behavior near the boundary of the function is first studied in a smooth domain and next studied in a Lipschitz domain or a doma with a still more general complicated boundary. In the half-space which is a smooth domain, with respect to two kinds of exclusion sets above-mentioned, the Winner type criteria and the behavior of superharmonic functions in the outside of the exceptional sets were researched by Essen etc. These results were extended to the results of the same kind near the infinite point of a corn which is the angle of the domain and also near the infinite point of the cylinder prolonged infinitely which is the cusp of the domain. Moreover, the results in the case of a coin and a cylinder were able to be obtained about another expression of the exceptional sets. These results were carried by the journal (Canadian Mathematical Bulletin) of Canada. It is decided to be carried by an American journal (Proc.Amer.Math.Soc. and Complex variables) and the journal (Czecho.Math.J.) of Czechoslovakia.