Perturbation results involving the 1-Laplace operator

Abstract

We consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.Keywords: 1-Laplace operator, eigenvalue problems, perturbation, nonsmooth critical point theory, weak slope * Both authors supported by DFG project "Variational problems related to the 1-Laplace operator".

A Note on Sharp Affine Poincaré-Sobolev Inequalities and Exact in Minimization of Zhang’s Energy on Bounded Variation and Exactness

“…This has mostly been in connection with the availability of solutions where the 1-Laplace operator is present, such as in the well-known Cheeger's problem [1]. See, among other places, [2]- [7] for contributions along this line of thought. A portion of them place a greater emphasis, more especially, on the challenge of reducing the functional The associated minimization issue involves proving the existence of minimizers for the least amount of energy.…”