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Some constructions with solutions of variable coefficient elliptic equations

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Abstract

We give a characterization of pairs of functionsf, g on the boundary of a compact manifold, which are the Dirichlet and Neumann boundary values for a solution of some second-order linear divergence-form elliptic equation, and we apply this to some other related questions in potential theory.

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References

  1. Greene, R., and Wu, H. Embeddings of open Riemannian manifolds by harmonic functions.Ann. Inst. Fourier (Grenoble) (1) 25, 215–235 (1975).

    MATH  MathSciNet  Google Scholar 

  2. Guillemin, V., and Pollack, A.Differential Topology. Prentice-Hall, Englewood Cliffs, NJ, 1974.

    MATH  Google Scholar 

  3. Hardt, R., and Simon, L. Nodal sets for solutions of elliptic equations.J. Diff. Geo. 30, 505–522 (1989).

    MATH  MathSciNet  Google Scholar 

  4. Hempel, J.3-Manifolds. Princeton University Press, Princeton, NJ, 1976.

    MATH  Google Scholar 

  5. Kenig, C. Carleman estimates, uniform Sobolev inequalities for second order differential operators, and unique continuation theorems.Proc. ICM, Berkeley, pp. 948–960 (1986), American Mathematical Society, 1987.

  6. Miller, K. Non-unique continuation for certain ODE’s in Hilbert space and for uniformly parabolic and elliptic equations in self-adjoint divergence form. InSymposium on Non-Well-Posed Problems and Logarithmic Convexity, R.J. Knops, ed., pp. 85–101. Springer Lecture Notes in Mathematics, Vol. 316, 1973.

  7. Milnor, J. A procedure for killing homotopy groups of differentiable manifolds.Proceedings of Symposia in Pure Mathematics 3, 39–55 (1961).

    MathSciNet  Google Scholar 

  8. Milnor, J.Lectures on the h-Cobordism Theorem. Princeton University Press, Princeton, NJ, 1965.

    MATH  Google Scholar 

  9. Ng, K.-F., and Wang, Y.-C.Partially Ordered Topological Vector Spaces. Oxford Mathematical Monographs. Clarendon Press, London, 1973.

    MATH  Google Scholar 

  10. Plis, A. On non-uniqueness in the Cauchy problem for an elliptic second order differential equation.Bull. Acad. Sci. Polon., Ser. Sci. Math. Astro. Phys. 11, 95–100 (1963).

    MATH  MathSciNet  Google Scholar 

  11. Sullivan, D. Cycles for the dynamical study of foliated manifolds and complex manifolds.Inventiones Math. 36, 225–255 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  12. Sullivan, D. A homological characterization of foliations consisting of minimal surfaces.Comment. Math. Helv. 54(2), 218–223 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  13. Sylvester, J., and Uhlmann, C. The Dirichlet to Neumann map and applications. InInverse Problems in Partial Differential Equations, D. Colton, R. Ewing, and W. Rundell, eds., pp. 99–139. SIAM, Philadelphia, 1990.

    Google Scholar 

  14. Wallace, A. H. Modifications and cobounding manifolds.Canadian J. Math. 12, 503–528 (1960).

    MathSciNet  Google Scholar 

  15. Wolff, T. Note on counterexamples in strong unique continuation problems.Proc. Amer. Math. Soc. (2) 114, 351–356 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  16. Wolff, T. Recent work on sharp estimates in second order elliptic unique continuation problems. Conference proceedings, Miraflores de la Sierra, 1992.J. Geom. Anal., to appear.

  17. Wolff, T. A counterexample in a unique continuation problem.Communications in Analysis and Geometry, to appear.

  18. Calabi, E. An intrinsic characterization of harmonic one forms. InGlobal Analysis: Papers in Honor of K. Kodaira, pp. 101–117, D. C. Spencer and S. Iyanaga, eds. Princeton University Press, Princeton, 1969.

    Google Scholar 

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  1. Department of Mathematics, University of California, 970 Evans Hall, 94720, Berkeley, CA

    Thomas H. Wolff

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  1. Thomas H. Wolff
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Wolff, T.H. Some constructions with solutions of variable coefficient elliptic equations. J Geom Anal 3, 423–511 (1993). https://doi.org/10.1007/BF02921289

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  • DOI: https://doi.org/10.1007/BF02921289

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