On Morrey Spaces in the Calculus of Variations

Authors

Kyle Fey, University of Nebraska-LincolnFollow

First Advisor

Mikil Foss

Date of this Version

5-2011

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Mikil Foss. Lincoln, Nebraska: May, 2011

Copyright 2011 Kyle Fey

Abstract

We prove some global Morrey regularity results for almost minimizers of functionals of the form u → ∫_Ω f(x, u, ∇u)dx. This regularity is valid up to the boundary, provided the boundary data are sufficiently regular. The main assumption on f is that for each x and u, the function f(x, u, ·) behaves asymptotically like the function h(|·|)^α(x), where h is an N-function.

Following this, we provide a characterization of the class of Young measures that can be generated by a sequence of functions {f_j} uniformly bounded in the Morrey space L^{p, λ}(Ω; R^N) with {|f_j|^p} equiintegrable. We then treat the case that each f_j = ∇u_j.

Lastly, we provide applications of and connections between these results.