Fractals in Engineering: Theoretical Aspects and Numerical Approximations

1. Front Matter

   Pages i-x

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2. A Numerical Approach to a Nonlinear Diffusion Model for Self-Organized Criticality Phenomena

   • C. Alberini, S. Finzi Vita

   Pages 1-25

3. Approximation of 3D Stokes Flows in Fractal Domains

   • Massimo Cefalo, Simone Creo, Mirko Gallo, Maria Rosaria Lancia, Paola Vernole

   Pages 27-53

4. ∞-Laplacian Obstacle Problems in Fractal Domains

   • Salvatore Fragapane

   Pages 55-77

5. Discretization of the Koch Snowflake Domain with Boundary and Interior Energies

   • Malcolm Gabbard, Carlos Lima, Gamal Mograby, Luke Rogers, Alexander Teplyaev

   Pages 79-102

6. On the Dimension of the Sierpinski Gasket in l ²

   • M. V. Marchi

   Pages 103-124

7. On the External Approximation of Sobolev Spaces by M-Convergence

   • Umberto Mosco, Maria Agostina Vivaldi

   Pages 125-154

8. Generalization of Rellich–Kondrachov Theorem and Trace Compactness for Fractal Boundaries

   • Anna Rozanova-Pierrat

   Pages 155-173

Fractal structures or geometries currently play a key role in all models for natural and industrial processes that exhibit the formation of rough surfaces and interfaces. Computer simulations, analytical theories and experiments have led to significant advances in modeling these phenomena across wild media. Many problems coming from engineering, physics or biology are characterized by both the presence of different temporal and spatial scales and the presence of contacts among different components through (irregular) interfaces that often connect media with different characteristics. This work is devoted to collecting new results on fractal applications in engineering from both theoretical and numerical perspectives. The book is addressed to researchers in the field.

• PDEs
• fractal analysis
• integral equations on fractals
• numerical approximation in fractal domains
• Theory and Applications in Engineering

• Dipartimento di Scienze di base ed Applicate per l’ingegneria, Sapienza Università di Roma, Roma, Italy

  Maria Rosaria Lancia

• Department of Mathematics, CentraleSupélec, Université Paris Saclay, Gif-sur-Yvette, France

  Anna Rozanova-Pierrat

Maria Rosaria Lancia is Professor of Mathematical Analysis at Sapienza University 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 of the J. of Applied Mathematics and Computation, Hill Publishing Group.

Anna Rozanova-Pierrat is Associate Professor of Applied Mathematics in CentraleSupélec, University Paris-Saclay, France. She obtained his PhD on Applied Mathematics in 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 (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.

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