Variational Methods for Polycrystal Plasticity and Related Topics in Partial Differential Equations

Abstract

In the first part of the thesis the effective yield set of ionic polycrystals is characterized by means of variational principles in L∞ that are associated to supremal functionals acting on matrix-valued divergence-free fields. The second part of the thesis is concerned with the study of the asymptotic behavior, as p → ∞, of the first and second eigenvalues and the corresponding eigenfunctions for the p(x)-Laplacian with Robin and Neumann boundary conditions, in an open, bounded domain with smooth boundary. We obtain uniform bounds for the sequences of eigenvalues (suitably rescaled), and we prove that the positive eigenfunctions converge uniformly to viscosity solutions of problems involving the ∞-Laplacian subject to appropriate boundary conditions.