Eigenvalue homogenization for the $p-$Laplacian with rapidly oscillating $L^q$ weights

In this article, we study the convergence rates of variational eigenvalues of the $p$-Laplacian with rapidly oscillating weights and potentials belonging in a suitable $L^q$ space. Our analysis covers both Dirichlet and Neumann boundary conditions, and we derive explicit estimates in terms of the eigenvalue index $k\in \mathbb{N}$ and the oscillation parameter $\epsilon >0$. These results are obtained through a detailed examination of certain oscillatory integrals and extend previously known results in the literature.