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SEMA SIMAI Springer Series ICIAM 2019 SEMA SIMAI Springer Series Volume 8 Editor-in-Chief Amadeu Delshams, Departament de Matemàtiques and Laboratory of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, Barcelona, Spain Series Editors Francesc Arandiga Llaudes, Departamento de Matemàtica Aplicada, Universitat de València, Valencia, Spain Macarena Gómez Mármol, Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain Francisco M. Guillén-González, Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain Francisco Ortegón Gallego, Departamento de Matemáticas, Facultad de Ciencias del Mar y Ambientales, Universidad de Cádiz, Puerto Real, Spain Carlos Parés Madroñal, Departamento Análisis Matemático, Estadística e I.O., Matemática Aplicada, Universidad de Málaga, Málaga, Spain Peregrina Quintela, Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Carlos Vázquez-Cendón, Department of Mathematics, Faculty of Informatics, Universidade da Coruña, A Coruña, Spain Sebastià Xambó-Descamps, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
This sub-series of the SEMA SIMAI Springer Series aims to publish some of the most relevant results presented at the ICIAM 2019 conference held in Valencia in July 2019. The sub-series is managed by an independent Editorial Board, and will include peer-reviewed content only, including the Invited Speakers volume as well as books resulting from mini-symposia and collateral workshops. The series is aimed at providing useful reference material to academic and researchers at an international level. More information about this subseries at http://www.springer.com/series/16499
Maria Rosaria Lancia • Anna Rozanova-Pierrat Editors Fractals in Engineering: Theoretical Aspects and Numerical Approximations
Editors Maria Rosaria Lancia Anna Rozanova-Pierrat Dipartimento di Scienze di base ed Department of Mathematics Applicate per l’ingegneria CentraleSupélec, Université Paris Saclay Sapienza Università di Roma Gif-sur-Yvette, France Roma, Italy ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISSN 2662-7183 ISSN 2662-7191 (electronic) ICIAM 2019 SEMA SIMAI Springer Series ISBN 978-3-030-61802-5 ISBN 978-3-030-61803-2 (eBook) https://doi.org/10.1007/978-3-030-61803-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Many natural and industrial processes lead to the formation of rough surfaces
and interfaces. Computer simulations, analytical theories and experiments led to
significant advances, in modeling these phenomena across wild media. Fractals
provide a good tool to describe such wild geometries, as well as in the case of those
phenomena that take place in small volumes and large surfaces.
This special issue is devoted to collect new results on fractal applications in
engineering from both a theoretical and a numerical point of view, having in mind
further developments of fractal and prefractal geometries in industrial applications.
Some of these results have been presented during the special Minisymposium
MS256 entitled Fractal Applications in Engineering: Theoretical Aspects and
Numerical Approximations, held in Valencia, Spain, July 15–19, 2019.
We thank all the authors for participating to this special issue as well as all
the referees who contributed with their constructive and rigorous reviews. Special
thanks go to the Editors-in-Chief Pablo Pedregal Tercero of University of Castilla-
La Mancha and Luca Formaggia of Politecnico di Milano for inviting us to edit a
Special Issue of the SEMA SIMAI Springer Series.
Guest Editors
Roma, Italy Maria Rosaria Lancia
Gif-sur-Yvette, France Anna Rozanova-Pierrat
vContents
A Numerical Approach to a Nonlinear Diffusion Model
for Self-Organized Criticality Phenomena . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
C. Alberini and S. Finzi Vita
Approximation of 3D Stokes Flows in Fractal Domains .. . . . . . . . . . . . . . . . . . . . 27
Massimo Cefalo, Simone Creo, Mirko Gallo, Maria Rosaria Lancia,
and Paola Vernole
∞-Laplacian Obstacle Problems in Fractal Domains. . . .. . . . . . . . . . . . . . . . . . . . 55
Salvatore Fragapane
Discretization of the Koch Snowflake Domain with Boundary
and Interior Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79
Malcolm Gabbard, Carlos Lima, Gamal Mograby, Luke Rogers,
and Alexander Teplyaev
On the Dimension of the Sierpinski Gasket in l 2 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103
M. V. Marchi
On the External Approximation of Sobolev Spaces by M-Convergence . . . 125
Umberto Mosco and Maria Agostina Vivaldi
Generalization of Rellich–Kondrachov Theorem and Trace
Compactness for Fractal Boundaries . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 155
Anna Rozanova-Pierrat
viiEditors and Contributors
About the Editors
Maria Rosaria Lancia is Professor of Mathematical Analysis at the Sapienza Uni-
versity of Rome, where she received her PhD in Applied and Theoretical Mechanics.
Her current research interests are fractal analysis and numerical approximation
of BVPs in fractal domains. The emphasis is on linear, quasilinear and fractional
BVPs in and within fractal domains possibly with dynamical boundary conditions
and vector analysis on fractafolds. She is an editorial board member of Fractal
and Fractional, MDPI, and she has been a member of the Journal of Applied
Mathematics and Computation, Hill Publishing Group up to 2019.
Anna Rozanova-Pierrat is Associate Professor of Applied Mathematics at the
CentraleSupélec, University Paris-Saclay, France. She obtained his PhD on Applied
Mathematics at the University Pierre et Marie Currie Paris 6 and RUDN (Moscow,
Russia), where she finished her studies on Theoretical and Applied Mathematics.
Her current research interests are motivated by physical and engineer problems
(such as models of non-linear acoustics, de Gennes hypothesis on the speed of the
heat propagation between two media, shape optimization) involving irregular and
fractal boundaries.
Contributors
C. Alberini Dipartimento SBAI, Sapienza Università di Roma, Rome, Italy
Massimo Cefalo Dipartimento di Ingegneria Informatica, Automatica e Ges-
tionale, Sapienza Università di Roma, Roma, Italy
Simone Creo Dipartimento di Scienze di Base e Applicate per l’Ingegneria,
Sapienza Università di Roma, Roma, Italy
ixx Editors and Contributors Salvatore Fragapane Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza” Università di Roma, Roma, Italy Malcolm Gabbard Colorado College, Department of Mathematics and Computer Science, Colorado Springs, CO, USA Mirko Gallo Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Roma, Italy Maria Rosaria Lancia Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Roma, Italy Carlos Lima California State University Long Beach, Department of Physics and Astronomy, Long Beach, CA, USA M. V. Marchi Dip. di Matematica, Sapienza Università di Roma, Roma, Italy Gamal Mograby University of Connecticut, Department of Mathematics, Storrs, CT, USA Umberto Mosco Department of Mathematics, Worcester Polytechnic Institute, Worcester, MA, USA Luke Rogers University of Connecticut, Department of Mathematics, Storrs, CT, USA Anna Rozanova-Pierrat CentraleSupélec, Université Paris-Saclay, Saint-Aubin, France Alexander Teplyaev University of Connecticut, Department of Mathematics, Storrs, CT, USA Paola Vernole Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Roma, Italy S. F. Vita Dipartimento di Matematica, Sapienza Università di Roma, Rome, Italy Maria Agostina Vivaldi Dip. Di Scienze di base ed Applicate per l’ingegneria, Sapienza Università di Roma, Roma, Italy
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