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Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

Christian Baer
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Abstract

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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Christian Baer. 1998-05-13. Zero Sets of Solutions to Semilinear Elliptic Systems of First Order. https://doi.org/10.1007/s002220050346

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