Measure upper bounds of nodal sets of Robin eigenfunctions

In this paper, we will establish the upper bounds of the Hausdorff measure of nodal sets of eigenfunctions with the Robin boundary conditions, i.e.,{Delta u + lambda u = 0, in Omega,u(nu) + mu u = 0, on partial derivative Omega,where the domain Omega subset of R-n, u(nu )is the derivative of u along the outer normal direction on partial derivative Omega. We will show that, if Omega is bounded and analytic, and the corresponding eigenvalue lambda is large enough, then the measure upper bounds for the nodal sets of eigenfunctions are C root lambda, where C is a positive constant depending only on n and Omega but not on mu. We also show that, if partial derivative Omega is C-infinity smooth and partial derivative Omega\Gamma is piecewise analytic, where Gamma subset of partial derivative Omega is a union of some n- 2 dimensional submanifolds of partial derivative Omega, mu > 0, and lambda is large enough, then the corresponding measure upper bounds for the nodal sets of u are C(root lambda + mu(alpha) + mu(-c alpha)) for any alpha is an element of (0, 1), where C is a positive constant depending on alpha, n, Omega and Gamma, and c is a positive constant depending only on n.