Lin's nodal and singular set measure conjecture for elliptic equations

Let $u$ solve the divergence-form elliptic equation

$$\partial_i(a^{ij}(x)\partial_j u)+b^i(x)\partial_i u+c(x)u=0$$

in $B(0,1)$, with

$$\Lambda^{-1}|\xi|^2\leq a^{ij}\xi_i\xi_j\leq\Lambda|\xi|^2,\qquad |b^i|,|c|\leq\Lambda,$$

and

$$|a^{ij}(x)-a^{ij}(y)|\leq\Lambda|x-y|.$$

Let $Z(u)$ be the nodal set, $S(u)$ the singular set, and $\beta_u$ the frequency quantity used in the source. Lin's conjecture. The nodal and singular sets satisfy

$$\mathcal{H}^{n-1}(Z(u)\cap B(0,1/2))\leq C(n,\Lambda)\beta_u(0,1)$$

and

$$\mathcal{H}^{n-2}(S(u)\cap B(0,1/2))\leq C(n,\Lambda)\beta_u^2(0,1).$$

These estimates concern quantitative control of nodal and singular sets for elliptic equations; the source does not state their resolution status.