Zero sets of solutions to semilinear elliptic systems of first order |
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Authors: | Christian Bär |
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Institution: | Mathematisches Insitut, Universit?t Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany (e-mail: baer@mathematik.uni-freiburg.de), DE
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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.
Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999 |
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Keywords: | Mathematics Subject Classification (1991): 35B05 |
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