A Variational Theory of Convolution-Type Functionals

Description

This book provides a general treatment of a class of functionals modelled on convolution energies with kernel having finite p-moments. A general asymptotic analysis of such non-local functionals is performed, via Gamma-convergence, in order to show that the limit may be a local functional representable as an integral. Energies of this form are encountered in many different contexts and the interest in building up a general theory is also motivated by the multiple interests in applications (e.g. peridynamics theory, population dynamics phenomena and data science). The results obtained are applied to periodic and stochastic homogenization, perforated domains, gradient flows, and point-clouds models. This book is mainly intended for mathematical analysts and applied mathematicians who are also interested in exploring further applications of the theory to pass from a non-local to a local description, both in static problems and in dynamic problems.   Preface Contents 1 Introduction References 2 Convolution-Type Energies 2.1 Notation 2.2 Setting of the Problem and Comments 2.3 Assumptions Reference 3 The -Limit of a Class of Reference Energies 3.1 The -Limit of G[a] References 4 Asymptotic Embedding and Compactness Results 4.1 An Extension Result 4.2 Control of Long-Range Interactions with Short-Range Interactions 4.3 Compactness in Lp Spaces 4.4 Poincaré Inequalities References 5 A Compactness and Integral-Representation Result 5.1 The Integral-Representation Theorem 5.2 Truncated-Range Functionals 5.3 Fundamental Estimates 5.4 Proof of the Integral-Representation Theorem 5.5 Convergence of Minimum Problems 5.6 Euler-Lagrange Equations 5.6.1 Regularity of Functionals F 5.6.2 Relations with Minimum Problems References 6 Periodic Homogenization 6.1 A Homogenization Theorem 6.2 The Convex Case 6.3 Relaxation of Convolution-Type Energies 6.4 An Extension Lemma from Periodic Lipschitz Domains 6.5 Homogenization on Perforated Domains References 7 A Generalization and Applications to Point Clouds 7.1 Perturbed Convolution-Type Functionals 7.2 Application to Functionals Defined on Point Clouds References 8 Stochastic Homogenization References 9 Application to Convex Gradient Flows 9.1 The Minimizing-Movement Approach to Gradient Flows 9.2 Homogenized Flows for Convex Energies References Index