Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

Details Zero Sets of Solutions to Semilinear Elliptic Systems of First Order Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

1998-05-13

arxiv.org/abs/math/9805065v1

Inventiones Math. 138 , 183-202 (1999)

Mathematics

Analysis of PDEs

16 pages, LaTeX2e, 2 figs, uses pstricks macro package

Scientific paper

10.1007/s002220050346

Consider a nontrivial 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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