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A unique continuation property on the boundary for solutions of elliptic equations


Author: Zhi Ren Jin
Journal: Trans. Amer. Math. Soc. 336 (1993), 639-653
MSC: Primary 31B20; Secondary 31B35, 35B60, 35J67
MathSciNet review: 1085944
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Abstract: We prove the following conclusion: if $ u$ is a harmonic function on a smooth domain $ \Omega $ in $ {R^n}$ , $ n \geq 3$ , or a solution of a general second-order linear elliptic equation on a domain $ \Omega $ in $ {R^2}$, and if there are $ {x_0} \in \partial \Omega $ and constants $ a$, $ b > 0$ such that $ \vert u(x)\vert \leq a\exp \{ - b/\vert x - {x_0}\vert\} $ for $ x \in \Omega $, $ \vert x - {x_0}\vert$ small, then $ u = 0$ in $ \Omega $ . The decay rate in our results is best possible by the example that $ u = $ real part of $ \exp \{ - 1/{z^\alpha }\} $ , $ 0 < \alpha < 1$ , is harmonic but not identically zero in the right complex half-plane.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1085944-3
Keywords: Unique continuation, solutions of elliptic equations
Article copyright: © Copyright 1993 American Mathematical Society