Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

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. Mathematics Subject Classification: 35B05