Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Comparison theorems for bounded solutions of $ \triangle u=Pu$


Author: Moses Glasner
Journal: Trans. Amer. Math. Soc. 202 (1975), 173-179
MSC: Primary 31C15; Secondary 30A48
DOI: https://doi.org/10.1090/S0002-9947-1975-0377088-X
MathSciNet review: 0377088
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ P$ and $ Q$ be $ {C^1}$ densities on a hyperbolic Riemann surface $ R$. A characterization of isomorphisms between the spaces of bounded solutions of $ \Delta u = Pu$ and $ \Delta u = Qu$ on $ R$ in terms of the Wiener harmonic boundary is given.


References [Enhancements On Off] (What's this?)

  • [1] M. Glasner and R. Katz, On the behavior of solutions of $ \Delta u = Pu$ at the Royden boundary, J. Analyse Math. 22 (1969), 343-354. MR 41 #1995. MR 0257344 (41:1995)
  • [2] A. Lahtinen, On the solutions of $ \Delta u = Pu$ for acceptable densities, Ann. Acad. Sci. Fenn. Ser. A I No. 515 (1972). MR 0419754 (54:7772)
  • [3] P. A. Loeb and B. Walsh, A maximal regular boundary for solutions of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 1, 283-308. MR 39 #4423. MR 0243099 (39:4423)
  • [4] M. Nakai, The space of bounded solutions of $ \Delta u = Pu$ on a Riemann surface, Proc. Japan Acad. 36 (1960), 267-272. MR 22 #12216. MR 0121478 (22:12216)
  • [5] -, Dirichlet finite solutions of $ \Delta u = Pu$ and classification of Riemann surfaces, Bull. Amer. Math. Soc. 77 (1971), 381-385. MR 45 #2162. MR 0293083 (45:2162)
  • [6] H. L. Royden, The equation $ \Delta u = Pu$ and the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I No. 271 (1959). MR 22 #12215. MR 0121477 (22:12215)
  • [7] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der math. Wissenschaften, Band 164, Springer-Verlag, Berlin and New York, 1970. MR 41 #8660. MR 0264064 (41:8660)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31C15, 30A48

Retrieve articles in all journals with MSC: 31C15, 30A48


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0377088-X
Keywords: Solution of $ \Delta u = Pu$, Riemann surfaces, Green's function, maximum principle, Wiener harmonic boundary
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society