Nodal sets for solutions of elliptic equations
Robert Hardt, Leon Simon · Journal of Differential Geometry · 1989
Here we study, on a connected domain ΩcR", the zero set u~ι{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~ι{0} is finite in a neighborhood of any point xo € Ω at which u has finite order of vanishing.(For Lipschitz ay this holds at each point XQ G Ω by the unique continuation theory for elliptic equations.)We actually obtain an explicit bound on the Hausdorff measure of u~ι {0} in terms of the order of vanishing of w, the modulus of continuity of α, 7 , and the bounds on α, 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 κ~!{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~ι{0} decomposes into a disjoint union of the embedded C 1 submanifold u~ι{0} Π {\Du\ > 0} together with the closed set u~ι{0} Π |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 Δw + /(x, u) = 0. We thank F. H. Lin for pointing out this reference.In §5 of the present paper we apply the main estimates of §1 and an estimate of Donnelly and Fefferman [2] for the order of vanishing of eigenfunctions to give an asymptotic bound of the (n -1 )-dimensional measure