Reflection and uniqueness theorems for harmonic functions

Armitage, D.

doi:10.1090/S0002-9939-99-04994-1

Reflection and uniqueness theorems for harmonic functions
HTML articles powered by AMS MathViewer

by D. H. Armitage PDF

Proc. Amer. Math. Soc. 128 (2000), 85-92 Request permission

Abstract:

Suppose that $h$ is harmonic on an open half-ball $\beta$ in $R^{N}$ such that the origin 0 is the centre of the flat part $\tau$ of the boundary $\partial \beta$. If $h$ has non-negative lower limit at each point of $\tau$ and $h$ tends to 0 sufficiently rapidly on the normal to $\tau$ at 0, then $h$ has a harmonic continuation by reflection across $\tau$. Under somewhat stronger hypotheses, the conclusion is that $h\equiv 0$. These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set $\tau$ can be replaced by a spherical surface, it cannot in general be replaced by a smooth $(N-1)$-dimensional manifold.

References

M. S. Baouendi and Linda Preiss Rothschild, A local Hopf lemma and unique continuation for harmonic functions, Internat. Math. Res. Notices 8 (1993), 245–251. MR 1233452, DOI 10.1155/S1073792893000273
M. S. Baouendi and L. P. Rothschild, Harmonic functions satisfying weighted sign conditions on the boundary, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1311–1318 (English, with English and French summaries). MR 1275201
M. Brelot, Éléments de la théorie classique du potentiel, “Les cours de Sorbonne”, vol. 3, Centre de Documentation Universitaire, Paris, 1965. 3e édition. MR 0178154
L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Ü. Kuran, Study of superharmonic functions in ${\bf {\rm }R}^{n}\times (0,\,+\infty )$ by a passage to ${\bf {\rm }R}^{n{\bf +3}}$, Proc. London Math. Soc. (3) 20 (1970), 276–302. MR 261023, DOI 10.1112/plms/s3-20.2.276