By Tibor Rado
A convex functionality f can be referred to as sublinear within the following experience; if a linear functionality l is ::=: j on the boundary issues of an period, then l:> j within the inside of that period additionally. If we change the phrases period and linear junction by means of the phrases area and harmonic functionality, we receive an announcement which expresses the attribute estate of subharmonic features of 2 or extra variables. This ge neralization, formulated and constructed through F. RIEsz, instantly at tracted the eye of many mathematicians, either as a result of its intrinsic curiosity and because of the big variety of its purposes. If f (z) is an analytic functionality of the complicated variable z = x + i y. then If (z) I is subharmonic. the possibility of a unfavorable mass-distribu tion is subharmonic. In differential geometry, surfaces of adverse curvature and minimum surfaces should be characterised by way of sub harmonic capabilities. the assumption of a subharmonic functionality ends up in major functions and interpretations within the fields simply pointed out, and· conversely, most of these fields is an it appears in exhaustible resource of recent theorems on subharmonic features, both by way of analogy or via direct implication.
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Extra info for On the Problem of Plateau / Subharmonic Functions
Example text
Choose a point Wo in D. Since cpdwo), cpz (wo), CPa (wo) are not all three equal to zero, suppose that CPa (wo) =l= 0, for instance. We restrict ourselves to a vicinity Do of w o , where CPa =l= o. From cpi + cP~ + cP~ = 0 it follows that (CPl - icpz) (CPl + icp2) = -cP~ . 26) Consequently CPl - icpz =l= 0 in Do. , # are analytic functions of w in Do, and # =l= 0,2 =l= 0 in Do . 2)#, CPl CPa = 2 A# • Conversely, if 2, # are analytic functions of w, then the functions CPl' CP2, CPa satisfy the condition cpi + cP~ + cP; = 0, as follows by direct computation.
Clearly CfJ~ + CfJ~ + CfJ; = O. Suppose there would exist, in the vicinity of w = 0, two single-valued analytic functions (]>, P such that 3w = (]>2_ p2, 5iw = i((]>2 P2), 4w = 2(]>P. 4w, and it is then clear that (]> is not singlevalued in the vicinity of w = O. 29), to have algebraic singularities. As far as the investigations to be reviewed in the sequel are concerned, this step has not been taken (d. Chapter V). 23. Since the purpose of this Chapter is to present only those facts about minimal surfaces which are significant from the point of view of the recent work on the problem of PLATEAU, we restrict ourselves to recall one more theorem, due to H.
See DARBOUX: Theorie generale des surfaces. Vol. 281-284. For the following formulas. see BOLZA: Vorlesungen iiber Variationsrechnung. 667. 1 2 107] II. Minimal surfaces in the small. 9. The preceding results have important implications. Suppose that the minimizing surface 5 is given in terms of isothermic parameters!. 7) reduce to xuu + xvv = °, + yvv = 0 , yuu Zuu + Zvv = 0. That is to say: if the minimizing surface 5 is given in terms of isothermic parameters, then the coordinate functions x (u, v), y (u, v), z (u, v) are harmonic functions.
