Extremal p -Laplacian eigenvalues

We study the shape optimization problem of variational Dirichlet and Neumann p -Laplacian eigenvalues, with area and perimeter constraints. We prove some results that characterize the optimizers and derive the formula for the Hadamard shape derivative of Neumann p -Laplacian eigenvalues. Then, we propose a numerical method based on the radial basis functions method to solve the eigenvalue problems associated to the p -Laplacian operator. Several numerical results are presented and some new conjectures are addressed.