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Remarks on the local Hopf's lemma
Author(s):
Vladimir
Shklover
Journal:
Proc. Amer. Math. Soc.
124
(1996),
2711-2716.
MSC (1991):
Primary 30B40;
Secondary 30C20, 30D50
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Abstract:
The paper deals with the problem of extending the recent work of M.S.Baouendi and L.P.Rothschild concerning harmonic functions vanishing to infinite order in the normal direction in balls and half-spaces. Contrary to what one expects, we show that the B.-R. result extends neither to arbitrary domains nor to cases when the normal is replaced by a curve transversal to the boundary. The exact criterion when the result holds in  is given.
References:
- [1]
- M.S.Baouendi and L.P.Rothschild, A local Hopf lemma and unique continuation for harmonic functions, Duke Math.J. 71, International Mathematics Research Notices 8 (1993), 245-251. MR 94i:31008
- [2]
- P.J.Davis, The Schwarz Function and its Applications, The Carus Mathematical Monographs, MAA, 1974. MR 53:11031
- [3]
- D.Khavinson, On reflection of harmonic functions in surfaces of revolution, Complex Variables 17 (1991), 7--14. MR 92j:31005
- [4]
- B.F.Logan,Jr., Properties of high-pass signals, Dissertation presented to the Electrical Engineering Faculty of Columbia University, New York, 1965.
- [5]
- H.S.Shapiro, Functions with a spectral gap, Bull.A.M.S. 79 (1973), 355--360. MR 49:7696
- [6]
- H.S.Shapiro, Notes on a Theorem of Baouendi and Rothschild, TRITA-MAT-1994-0022, Royal Inst. of Tech., Stockholm, Sweden. CMP 95:17
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Additional Information:
Vladimir
Shklover
Affiliation:
Department of Mathematical Sciences, University
of Arkansas, Fayetteville,
Arkansas 72701
Address at time of publication:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email:
shklover@wam.umd.edu
DOI:
10.1090/S0002-9939-96-03367-9
PII:
S 0002-9939(96)03367-9
Keywords:
Harmonic functions,
Hopf's lemma,
analytic continuation
Received by editor(s):
July 6, 1994
Received by editor(s) in revised form:
February 28, 1995
Communicated by:
J. Marshall Ash
Copyright of article:
Copyright
1996,
American Mathematical Society
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