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Inventiones mathematicae

, Volume 138, Issue 1, pp 183–202 | Cite as

Zero sets of solutions to semilinear elliptic systems of first order

  • Christian Bär
Article

Abstract.

Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.

Mathematics Subject Classification (1991): 35B05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Christian Bär
    • 1
  1. 1.Mathematisches Insitut, Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany (e-mail: baer@mathematik.uni-freiburg.de)Germany

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