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Boundary unique continuation theorems under zero Neumann boundary conditions

Published online by Cambridge University Press:  17 April 2009

Xiangxing Tao  and
Songyan Zhang
Xiangxing Tao
Affiliation:
Department of Mathematics, Faculty of Science, Ningbo University, Ningbo, 315211, Peoples Republic of China, e-mail: [email protected]
Songyan Zhang
Affiliation:
Department of Mathematics, Faculty of Science, Ningbo University, Ningbo, 315211, Peoples Republic of China, e-mail: [email protected]
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Let u be a solution to a second order elliptic equation with singular potentials belonging to the Kato-Fefferman-Phong's class in Lipschitz domains. We prove the boundary unique continuation theorems and the doubling properties for u2 near the boundary under the zero Neumann boundary condition.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 1 , August 2005 , pp. 67 - 85
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Adolfsson, V. and Escauriaza, L., ‘C 1,α domains and unique continuation at the boundary’, Comm. Pure Appl. Math. 50 (1997), 935–969.3.0.CO;2-H>CrossRefGoogle Scholar
[2]Adolfsson, V., Escauriaza, L. and Kenig, C., ‘Convex domains and unique continuation at the boundary’, Rev. Mat. Iberoamericana 11 (1995), 513–1525.CrossRefGoogle Scholar
[3]Aizenman, M. and Simon, B., ‘Brownian motion and Harnack's inequality for Schrödinger operators’, Comm. pure Appl. Math. 35 (1982), 209–273.CrossRefGoogle Scholar
[4]Chiarenza, F., Fabes, E. B. and Garofalo, N., ‘Harnack's inequality for Schrödinger operators and the continuity of solutions’, Proc. Amer. Soc. 98 (1986), 415–425.Google Scholar
[5]Fabes, E., Garofalo, N. and Lin, F-H., ‘A partial answer to a conjecture of B. Simon concerning unique continuation’, J. Funct. Anal. 88 (1990), 194–210.CrossRefGoogle Scholar
[6]Garofalo, N. and Lin, F-H., ‘Monotonicity properties of variational integrals, Ap weights and unique continuation’, Indiana Univ. Math. J. 35 (1986), 245–268.CrossRefGoogle Scholar
[7]Garofalo, N. and Lin, F-H., ‘Unique continuation for elliptic operators: a geometric-variational approach’, Comm. Pure Appl. Math. 40 (1987), 347–366.CrossRefGoogle Scholar
[8]Kenig, C.E., Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics 83(Amercan Mathematical Society,Providence R.I.,1994).CrossRefGoogle Scholar
[9]Kurata, K., ‘A unique continuation theorem for uniformly elliptic equations with strongly singular potentials’, Comm. Partial Differential Equations 18 (993), 1161–1189.CrossRefGoogle Scholar
[10]Kurata, K., ‘A unique continuation theorem for the Schrödinger equation with singular magnetic field’, Proc. Amer. Soc. Math. 125 (1997), 853–860.CrossRefGoogle Scholar
[11]Simon, B., ‘Schrödinger semigroups’, Bull. Amer. Math. Soc. 7 (1982), 447–521.CrossRefGoogle Scholar
[12]Tao, X.X., ‘Doubling properties and unique continuation at the boundary for elliptic operators with singular magnetic fields’, Studia Math. 151 (2002), 31–48.CrossRefGoogle Scholar
[13]Verchota, G., ‘Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains’, J. Funct. Anal. 59 (1984), 572–611.CrossRefGoogle Scholar