Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

Autoren: Christian Bär (1999)

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.

Invent. Math.
138, no. 1

zur Übersicht der Publikationen