Nodal sets for solutions of elliptic equations 论文

摘要

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} {\Du\ > 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 dimw^O} 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.