Non-unique solutions to boundary value problems for non-symmetric divergence form equations
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- by Andreas Axelsson PDF
- Trans. Amer. Math. Soc. 362 (2010), 661-672 Request permission
Abstract:
We calculate explicitly solutions to the Dirichlet and Neumann boundary value problems in the upper half plane, for a family of divergence form equations having non-symmetric coefficients with a jump discontinuity. It is shown that the boundary equation method and the Lax–Milgram method for constructing solutions may give two different solutions when the coefficients are sufficiently non-symmetric.References
- Alfonseca, M., Auscher, P., Axelsson, A., Hofmann, S., and Kim, S. Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^\infty$ coefficients. Preprint.
- Auscher, P., Axelsson, A., and Hofmann, S. Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. Preprint.
- C. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), no. 2, 231–298. MR 1770930, DOI 10.1006/aima.1999.1899
- Kenig, C., and Rule, D. The regularity and Neumann problem for non-symmetric elliptic operators. Preprint.
- M$^\text {c}$Intosh, A., and Qian, T. Convolution singular integral operators on Lipschitz curves. In Harmonic analysis (Tianjin, 1988), vol. 1494 of Lecture Notes in Mathematics Springer, Berlin, 1991, pp. 142–162.
Additional Information
- Andreas Axelsson
- Affiliation: Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden
- Email: andax@math.su.se
- Received by editor(s): September 14, 2007
- Published electronically: September 18, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 661-672
- MSC (2000): Primary 35J25, 42A50
- DOI: https://doi.org/10.1090/S0002-9947-09-04673-X
- MathSciNet review: 2551501