Stiffness of harmonic functions
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- by Cecilia Y. Wang PDF
- Proc. Amer. Math. Soc. 77 (1979), 103-106 Request permission
Abstract:
Harmonic functions cannot change rapidly. For example, if K is a compact subset of a Riemann surface R and {u} a family of harmonic functions u on R of nonconstant sign on K, then it is known that there exists a constant $q \in (0,1)$ independent of u such that ${\max _K}|u| \leqslant q{\sup _R}|u|$ for all $u \in \{ u\}$. In the present note we shall show that relations expressing such “stiffness” of harmonic functions can also be given for the Dirichlet norm and for the partial derivative with respect to the Green’s function.References
- Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
- Mitsuru Nakai, On Evans potential, Proc. Japan Acad. 38 (1962), 624–629. MR 150296
- Kôtaro Oikawa, A constant related to harmonic functions, Jpn. J. Math. 29 (1959), 111–113. MR 120367, DOI 10.4099/jjm1924.29.0_{1}11
- Kôtaro Oikawa, Sario’s lemma on harmonic functions, Proc. Amer. Math. Soc. 11 (1960), 425–428. MR 114913, DOI 10.1090/S0002-9939-1960-0114913-6
- Burton Rodin and Leo Sario, Convergence of normal operators, K\B{o}dai Math. Sem. Rep. 19 (1967), 165–173. MR 218556
- Leo Sario, A linear operator method on arbitrary Riemann surfaces, Trans. Amer. Math. Soc. 72 (1952), 281–295. MR 46442, DOI 10.1090/S0002-9947-1952-0046442-2
- Leo Sario, An integral equation and a general existence theorem for harmonic functions, Comment. Math. Helv. 38 (1964), 284–292. MR 165097, DOI 10.1007/BF02566917
- Leo Sario, Classification of locally Euclidean spaces, Nagoya Math. J. 25 (1965), 87–111. MR 210885
- Leo Sario, Menahem Schiffer, and Moses Glasner, The span and principal functions in Riemannian spaces, J. Analyse Math. 15 (1965), 115–134. MR 184182, DOI 10.1007/BF02787691
- Leo Sario and Georges Weill, Normal operators and uniformly elliptic self-adjoint partial differential equations, Trans. Amer. Math. Soc. 120 (1965), 225–235. MR 192175, DOI 10.1090/S0002-9947-1965-0192175-0
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 77 (1979), 103-106
- MSC: Primary 30F15
- DOI: https://doi.org/10.1090/S0002-9939-1979-0539639-0
- MathSciNet review: 539639