For additional information and updates on this book, visit www.ams.org/bookpages/gsm-105 Library of Congress Cataloging-in-Publication Data Leoni, Giovanni, 1967-A first course in Sobolev spaces / Giovanni Leoni. p. cm.-(Graduate studies in mathematics ; v. 105) Includes bibliographical references and index. ISBN 978-0-8218-4768-8 (alk. paper) 1. Sobolev spaces. I. Title. QA323.L46 2009 515 .782-dc22 2009007620 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

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Abstract. We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations.We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem.The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the -limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the formwhere β e represents the elastic part of the macroscopic strain, and Curl β e represents the geometrically necessary dislocation density. The plastic energy density ϕ is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.

Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavourable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup where the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero contact-angle condition between islands and wetting layers.

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