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(iARDEN 



AN EXAMPLE IN POTENTIAL THEORY. 

 By O. D. Kellogg. 



Received Feb. 21, 1923. Presented May 9, 1923. 



1. The First Boundary Value Problem of Potential 



Theory. 



Given an open continuum, T, in the plane of x and y, and values, F, 

 assigned to each of the boundary points of T, the first boundary value 

 problem of potential theory consists in finding a function which shall 

 have continuous second derivatives which satisfy LaPIace's equation 

 at the points of T, and which shall approach the given values, F, on 

 the boundary. For continuous boundary values, and for a broad 

 class of regions T, the problem has been proven possible of solution. 1 

 We shall denote by [7i] the class of regions for which the problem is 

 possible in this sense. 



2. The Example. 



The purpose of the present note is to give an example of a region T\ 

 to which previous proofs do not apply; and incidentally, to throw 

 some light on the direction which further investigation may take. 



We first recall a familiar point set obtained by the "removal of 



middle thirds." The set So is the closed interval f — -, - J of the .r-axis. 

 The set Si is obtained from So by the removal of its open middle third, 

 and consists, therefore, in the two closed intervals (—-,—-] arid 



1 The most general results of this class appear to be due to Lebesgue, Sur Ie 

 probleme de Dirichlet, Rendiconti di Palermo, v. 24 (1907), pp. 371-402. His 

 restriction on T is, that any point belongs to T if it is possible to surround the 

 point by a curve, lying in an arbitrarily small neighborhood of the point, and 

 consisting only of points of T. Results more general, at least in some direc- 

 tions, have been obtained by H. B. Phillips and Norbert Wiener, and by 

 G. E. Raynor. Their papers have appeared, and are expected to appear, in the 

 Journal of Mathematics and Physics of the Massachusetts Institute of Technology, 

 vol. 2 (1923), pp. 105-124, and in the Annals of Mathematics, respectively. 

 For simply connected regions in particular, G. C. Evans has shown that the 

 problem is solvable when the values F are merely bounded summable func- 

 tions of the values on the boundary of the conjugate to a Green function, pro- 

 vided a suitable understanding be agreed on as to the manner of approach to 

 the boundary values. See Problems of Potential Theory, Proceedings of the 

 National Academy of Sciences, vol. 7 (1921), pp. 89-98. 



