&Z(u)\equiv \{x\in B_{1/2}(0):u(x)=0\}\, ,\notag\\
&Z_r(u)\equiv {\left\{x\in B_{1/2}(0): \inf_{B_r(x)}|u|^2(x)<\epsilon(n)\fint_{\partial B_{2r}(x)}|u|^2\right\}}\, .\end{aligned}$$ As with the effective critical and singular sets, the effective nodal set represents the set of points where $u$ has a definite size on a ball of definite size. It is again the frequency which plays a key role in controlling the nodal set, though in this case it is the singular frequency $N^u_{\mathcal{S}}$. The primary conjecture in the area goes back to Yau, which predicts that there exists a constant $C(n,\lambda)>0$ such that $$\begin{aligned}
H^{n-1}(Z(u))< C\Lambda\, .\end{aligned}$$ Yau’s conjecture has been proven in [@DonFef] for analytic coefficients. For Lipschitz coefficients the best result known are given in [@HardtSimon] which give the estimate $$\begin{aligned}
H^{n-1}(Z(u))< \Lambda^{C\Lambda}\, .\end{aligned}$$ The techniques of this paper, which are quite different from that of [@HardtSimon], can recapture that result, as well as improve it to the stronger Minkowski version. More specifically, our main theorem in this direction is the following:

\[t:main\_nodal\] Let $u:B_1(0)\to {\mathds{R}}$ solve (\[eq\_Lu\]) with the coefficients satisfying (\[e:coefficient\_estimates\]). There exists $r_0=r_0(n,\lambda)>0$ and $C=C(n,\lambda)$ such that if $\Lambda\equiv N^u(0,2r_0)$, then the following Minkowski estimates hold: $$\begin{aligned}
\label{e:mink_est_nod}
{\operatorname{Vol}}(B_r(Z(u))\cap B_{r_0}(0)) \leq {\operatorname{Vol}}(B_r(Z_r(u))\cap B_{r_0}(0)) \leq C^{\Lambda^2}\, r^2\, .\end{aligned}$$ In particular, we have the much weaker 