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$ PD$-minimal solutions of $ \Delta u=Pu$ on open Riemann surfaces


Author: Wellington H. Ow
Journal: Proc. Amer. Math. Soc. 37 (1973), 85-91
MSC: Primary 30A48
DOI: https://doi.org/10.1090/S0002-9939-1973-0310233-4
MathSciNet review: 0310233
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Abstract: By means of the Royden compactification of an open Riemann surface R necessary and sufficient conditions are given for a Dirichlet-finite solution of $ \Delta u = Pu\;(P \geqq 0, P\;{\nequiv}\;0)$ to be PD-minimal on R. A relation between PD-minimal solutions and HD-minimal solutions is obtained. In addition it is shown that the dimension of the space of PD-solutions is the same as the number of P-energy nondensity points in the finite dimensional case.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310233-4
Keywords: Royden harmonic boundary, P-energy nondensity point, harmonic projection, PD-minimal function, HD-minimal function, Riesz decomposition
Article copyright: © Copyright 1973 American Mathematical Society

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