Zero sets of solutions to semilinear elliptic systems of first order

被引:44
作者
Bär, C [1 ]
机构
[1] Univ Freiburg, Math Inst, D-79104 Freiburg, Germany
关键词
D O I
10.1007/s002220050346
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
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页码:183 / 202
页数:20
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