Lower bounds for interior nodal sets of Steklov eigenfunctions (Q2821733)

Consider the following Steklov eigenvalue problem on a \(n\)-dimensional compact Riemannian manifold \(M\) with boundary \(\partial M\): NEWLINE\[NEWLINE\Delta e_\lambda =0\;\text{in}\;M,\;\;\partial_v e_\lambda =\lambda e_\lambda\;\;\text{on}\;\partial M,NEWLINE\]NEWLINE where \(v\) is the outward unit normal vector field of the boundary. For any eigenvalue \(\lambda>0\), consider the nodal set of the eigenfunction \(Z_\lambda:= \{e_\lambda =0\}\). By using a Dong type identity and gradient estimates of the eigenfunctions, the estimate NEWLINE\[NEWLINE|Z_\lambda|\geq c\lambda^{\frac{2-n}2}NEWLINE\]NEWLINE is proved for some constant \(c>0\) and all eigenvalues \(\lambda\), where \(|\cdot|\) is the \((n-1)\)-dimensional Hausdorff measure.