Lin’s singular-set measure conjecture for elliptic equations

Prove that every nontrivial solution of a divergence-form elliptic equation with Lipschitz coefficients satisfies the quantitative singular-set estimate \(\mathcal{H}^{n-2}(S(u)\cap B_{1/2}(0))\leq C N(0,1)^2\), where \(S(u)=\{u=0\}\cap\{\nabla u=0\}\), \(N(0,1)\) is the frequency function on \(B_1(0)\), and \(C\) is universal.

Background

The paper identifies a central conjecture concerning the quantitative size of singular sets of solutions to elliptic equations. The singular set is the intersection of the nodal set and the critical set, and its expected $(n-2)$-dimensional measure is conjectured to be controlled quadratically by the frequency at the reference scale.

The authors review partial results: finiteness of the relevant Hausdorff measure under smooth coefficients and weak or Minkowski-type estimates under Lipschitz or weaker coefficient assumptions. The paper develops estimates for nodal and critical sets under Dini-type hypotheses, but the quoted conjectural estimate is presented as a broader unresolved statement.