Applied Mathematical Problems in Geophysics: Cetraro, Italy 2019 (Lecture Notes in Mathematics, 2308)

✍ Scribed by Massimo Chiappini (editor), Vincenzo Vespri (editor)

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 211
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This CIME Series book provides mathematical and simulation tools to help resolve environmental hazard and security-related issues.

The contributions reflect five major topics identified by the SIES (Strategic Initiatives for the Environment and Security) as having significant societal impact: optimal control in waste management, in particular the degradation of organic waste by an aerobic biomass, by means of a mathematical model; recent developments in the mathematical analysis of subwave resonators; conservation laws in continuum mechanics, including an elaboration on the notion of weak solutions and issues related to entropy criteria; the applications of variational methods to 1-dimensional boundary value problems, in particular to light ray-tracing in ionospheric physics; and the mathematical modelling of potential electromagnetic co-seismic events associated to large earthquakes.
This material will provide a sound foundation for those who intend to approach similar problems from a multidisciplinary perspective.

✦ Table of Contents

Contents
1 Introduction to the CIME Series Volume
2 Optimal Control Strategies for Composting Processes in Biocells with L1-Type and L2-Type Cost Objectives
2.1 Introduction
2.2 The Optimal Control Problem
2.2.1 Formulation of the Problem
2.2.2 Analysis of the OC Problem
2.3 Numerical Results
2.3.1 Comparison of Strategies for Linear and Quadratic Cost Terms
2.3.2 Analysis for Varying Balance Between Costs
2.4 Conclusions
References
3 Wave Interaction with Subwavelength Resonators
3.1 Introduction
3.2 Subwavelength Resonances
3.2.1 Problem Setting
3.2.2 Layer Potential Theory on Bounded Domains and Gohberg-Sigal Theory
Layer Potential Operators
Generalized Argument Principle and Generalized Rouché's Theorem
3.2.3 Capacitance Matrix Analysis
3.3 Close-to-Touching Subwavelength Resonators
3.3.1 Coordinate System
3.3.2 Resonant Frequency Hybridization and Gradient Blow-Up
3.4 Effective Medium Theory for Subwavelength Resonators
3.4.1 High Refractive Index Effective Media
3.4.2 Double-Negative Metamaterials
3.5 Periodic Structures of Subwavelength Resonators
3.5.1 Floquet Theory
3.5.2 Quasi-Periodic Layer Potentials
3.5.3 Square Lattice Subwavelength Resonator Crystal
3.5.4 Subwavelength Band Gaps and Bloch Modes
Asymptotic Behavior of the First Bloch Eigenfrequency ω1α
3.6 Topological Metamaterials
3.6.1 Dirac Singularity
3.6.2 Topologically Protected Edge Modes
Sensitivity to Geometric Imperfections
Robustness Properties of One-Dimensional Chains of Subwavelength Resonators with Respect to Imperfections
3.7 Mimicking the Cochlea with an Array of Graded Subwavelength Resonators
3.8 Concluding Remarks
References
4 Variational Methods with Application to One-Dimensional Boundary Value Problems and Numerical Evaluations
4.1 Introduction
4.2 Setting of the Problem
4.3 Nonsmooth Critical Point Theory
4.4 Compactness and Lower Semicontinuity
4.5 The Variational Approach
4.6 A Coercive Case
4.7 Fermat's Principle and Numerical Evaluations for Light Rays
References
5 Electromagnetic Hypogene Co-seismic Sources
5.1 Introduction
5.1.1 Magnetic Anomalies of Possible Coseismic Nature
5.1.2 General Electrodynamic Models in Seismology
5.1.3 Constitutive Properties of the Propagation Medium
5.1.4 Grounds for Magneto-Quasistatic Models
5.1.5 Boundary Conditions
5.1.6 Parabolic Inverse Source Problems
5.2 Mathematical Framework
5.2.1 Energy Space
5.2.2 Time-Dependent Spaces
5.2.3 Regularity of the Boundaries
5.2.4 Assumptions on the Coefficients
5.2.5 Total Basis and Magnetic Eigenvalues
5.3 Well-Posedness for the Forward Problem
5.3.1 Parabolic Estimates
5.4 Hyperbolic Estimates and Their Singular Limit
5.5 A Forward Model with Singular Sources
5.5.1 Subsurface Sources
5.5.2 Signals in Free Homogeneous Space
5.5.3 Renormalised Signals
5.6 Inverse Source Problems
5.6.1 Nonuniqueness of Volume Currents
5.6.2 Uniqueness for Atomic Sources
5.6.3 Uniqueness of Subsurface Currents
5.6.4 Inverse Source Problems with Controllability
References
6 Conservation Laws in Continuum Mechanics
6.1 Introduction
6.2 Entropy Solutions
6.2.1 Weak Solutions
6.2.2 Rankine-Hugoniot Condition
6.2.3 Nonuniqueness of Weak Solutions
6.2.4 Entropy Conditions
6.2.5 Entropic Shocks
6.2.6 Change of Coordinates
6.2.7 Uniqueness and Stability of Entropy Solutions
6.3 Riemann Problem
6.3.1 Strictly Convex Fluxes
6.3.2 General Fluxes
6.4 Vanishing Viscosity
6.4.1 A Priori Estimates, Compactness, and Convergence
6.4.2 Error Estimate
Appendix: BV Functions
References