Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

Our topological setting is a smooth compact manifold of dimension two or higher with boundary. Although this underlying structure smooth, the Riemannian metric tensor only assumed to be bounded and measurable. This known as rough manifold. For large class boundary conditions we demonstrate Weyl law for asymptotics eigenvalues Laplacian associated metric. Moreover, obtain eigenvalue weighted Laplace equations Of particular novelty that weight function not fixed sign, thus may both positive negative. Key ingredients in proofs were demonstrated by Birman Solomjak nearly fifty years ago their seminal work on asymptotics. In addition determining equations, also wish promote achievements which have further applications modern problems.