Harmonic majorants of subharmonic functions
Throughout this Chapter, u will denote a function which is subharmonic in a domain G. Consider a region G′ + B′ comprised in G and a function H which is continuous in G′ + B′ and harmonic in G′. If H ≦ u oh B′, then H ≦ u in G′ also, by the definition of a subharmonic function. Naturally, one will try to use a harmonic majorant H which is as small as possible. Suppose that G′ + B′ is a Dirichlet region and suppose also that u is continuous. The solution H of the Dirichlet problem for G′ with the boundary condition H = u on B′ is then obviously the best harmonic majorant in G′. If however u is not continuous, then it is not clear that there exists a harmonic majorant in G′ which should be considered the best. This situation lead to investigations which will be reviewed presently.
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