This book contains some of the results presented at the mini-symposium titled Emerging Problems in the Homogenization of Partial Differential Equations, held during the ICIAM2019 conference in Valencia in July 2019. The papers cover a large range of topics, problems with weak regularity data involving renormalized solutions, eigenvalue problems for complicated shapes of the domain, homogenization of partial differential problems with strongly alternating boundary conditions of Robin type with large parameters, multiscale analysis of the potential action along a neuron with a myelinated axon, and multi-scale model of magnetorheological suspensions. The volume is addressed to scientists who deal with complex systems that presents several elements (characteristics, constituents...) of very different scales, very heterogeneous, and search for homogenized models providing an effective (macroscopic) description of their behaviors.

Autorentext

Patrizia Donato,** **Professor at the University of Rouen Normandie, is author of 3 books and about 100 international articles on Partial Differential Equations, in particular their Homogenization.  She has given about 100 lectures and seminars and 15 research courses in several countries, and organized about 20 international scientific events. She directed or co-directed 15 PhD thesis. Member of the EMS Ethics Commettee, and of the editorial board of Asymptotic Analysis, and Ricerche di Matematica. Manuel Luna-Laynez is Professor at the University of Seville. He is currently the Director of the Department of Differential Equations and Numerical Analysis. His main research interests concern homogenization theory and asymptotic analysis of partial differential equations posed in thin domains. He is the author of numerous publications focused on applications to the optimal design of materials, behavior of fluids in domains with rough boundaries, porous media and elastic multi-structures.

Inhalt
Nika, G. and Vernescu, B., Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids.- Jerez-Hanckes, C. et al., Multiscale analysis of myelinated axons.- Pérez-Martínez, M., Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions.-  G. Fulgencio, R. and Guibé, O.,  Quasilinear Elliptic Problems in a Two-Component Domain with L^1 data.- Donato, P. et al., Homogenization of  an eigenvalue problem in a two-component domain with  interfacial jump.