Abstract
surface 3x=h-'(X) and D(u; X) for the Dirichlet integral over the region ix bounded by a and fi. The main result of this paper is the inequality: maxhJ|Qx=m(h; X)=D(h; X)<m(u; X)<D(u; X), for uEEHo(Q), u?? h. Thus h minimizes the mean and the Dirichlet integral in the class Ho(Q). This inequality can be used in the classification of locally Euclidean spaces. In particular, we shall show that OG- OHOM = OHOD = OHOB. Here OG is the class of spaces V possessing no Green's function, and OHOK, K = M, D, B is the class of V on which there exist no nonconstant HoK functions, that is, harmonic functions which vanish on the border of a boundary neighborhood of V and which are of finite mean, of finite Dirichlet integral, or bounded, respectively.
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