Many problems in image analysis, digital processing and shape optimization can be expressed as variational problems involving the discretization of the Laplace-Beltrami operator. Such discretizations have have been widely studied for meshes or polyhedral surfaces. On digital surfaces, direct applications of classical operators are usually not satisfactory (lack of multigrid convergence, lack of precision.. .). In this paper, we first evaluate previous alternatives and propose a new digital Laplace-Beltrami operator showing interesting properties. This new operator adapts Belkin et al. [1] to digital surfaces embedded in 3D. The core of the method relies on an accurate estimation of measures associated to digital surface elements. We experimentally evaluate the interest of this operator for digital geometry processing tasks.