Nodal sets for solutions of elliptic equations

Here we study, on a connected domain cR", the zero set u~0 of a solution u of an elliptic equation aijDjDjU + bjDjU + cu = 0, where aij, bj, c are bounded and # /7 is continuous. Our principal result (precisely stated in Theorem (1. 7) below) is that the (n -l) -dimensional Hausdorff measure of u~0 is finite in a neighborhood of any point xo at which u has finite order of vanishing. (For Lipschitz ay this holds at each point XQ G by the unique continuation theory for elliptic equations. ) We actually obtain an explicit bound on the Hausdorff measure of u~ 0 in terms of the order of vanishing of w, the modulus of continuity of, 7, and the bounds on, j? , bj 9 c. Notice that in the case the coefficients ciij, bj, c are analytic, u is then real analytic 8, and the finiteness of the (n -l) -dimensional Hausdorff measure of ~!0 is automatic 3, 3. 4. 8. The explicit bound on the (n -1) -dimensional Hausdorff measure is nevertheless of interest in this case, but a more precise estimate for the real analytic case was already established in 2. We also show here (in Theorem (1. 10) ) that if the coefficients are sufficiently smooth then u~0 decomposes into a disjoint union of the embedded C 1 submanifold u~0 \ > 0 together with the closed set u~0 |Dw|~* (), which we show is countably (n -2) -rectifiable. L. Caffarelli and A. Friedman showed already in 1 that dimwO} n IDwl" 1 ^} < n -2 in the case of equations of the special form w + / (x, u) = 0. We thank F. H. Lin for pointing out this reference.