On Topologies and Boundaries in Potential Theory by Marcel Brelot download in pdf, ePub, iPad
As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities. Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the Kelvin transform and the method of images. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups such as the group of rotations or translations. There are results which describe the local structure of level sets of harmonic functions.
Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions.
Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series. More complicated situations can also happen. The most common instance of such a construction is to relate harmonic functions on a disk to harmonic functions on a half-plane. Third, one can use conformal transforms to map harmonic functions in one domain to harmonic functions in another domain. These convergence theorems are used to prove the existence of harmonic functions with particular properties.
This means that the fundamental object of study in potential theory is a linear space of functions. One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. This fact has several implications. This observation will prove especially important when we consider function space approaches to the subject in a later section.
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