Harmonic Analysis and Partial Differential Equations

✍ Scribed by Michael Ruzhansky, Jens Wirth

• Publisher: Birkhäuser
• Year: 2023
• Tongue: English
• Leaves: 241
• Series: Trends in Mathematics
• Category: Library

✦ Synopsis

This book collects papers related to the session “Harmonic Analysis and Partial Differential Equations” held at the 13th International ISAAC Congress in Ghent and provides an overview on recent trends and advances in the interplay between harmonic analysis and partial differential equations. The book can serve as useful source of information for mathematicians, scientists and engineers.
The volume contains contributions of authors from a variety of countries on a wide range of active research areas covering different aspects of partial differential equations interacting with harmonic analysis and provides a state-of-the-art overview over ongoing research in the field. It shows original research in full detail allowing researchers as well as students to grasp new aspects and broaden their understanding of the area.

✦ Table of Contents

Preface
Contents
List of Contributors
The Wave Resolvent for Compactly Supported Perturbations of Minkowski Space
1 Introduction
1.1 Motivation
1.2 Main Result and Sketch of Proof
1.3 Structure of Paper
2 Essential Self-Adjointness
2.1 Preliminaries on Self-Adjointness
2.2 Preliminaries on Microlocal Analysis
2.3 Proof of Local Regularity
2.4 Generalization to Static Spacetimes
3 Uniform Microlocal Estimates
3.1 Uniform Wavefront Set
3.2 Uniform Resolvent Estimate
References
Smoothing Effect and Strichartz Estimates for Some Time-Degenerate Schrödinger Equations
1 Introduction
2 Smoothing Effect and Local Well-Posedness for the Class Lα,c
2.1 The Class Lα
2.2 The Class Lα,c
3 Strichartz Estimates and Local Well-Posedness for Lb
Appendix
References
On the Cauchy Problem for the Nonlinear Wave Equation with Damping and Potential
1 Introduction
2 Formulation of the Problem and Results
3 Preliminaries
4 Proof of Theorem 1
5 Proof of Theorem 2
References
Local Well-Posedness for the Scale-Critical Semilinear Heat Equation with a Weighted Gradient Term
1 Introduction and Main Results
2 Proofs of Theorems 1.2 and 1.3
References
On the Rellich Type Inequality for Schrödinger Operators with Singular Potential
1 Introduction
2 Preliminary Observations
3 Proof of Theorem 1
References
Global Solutions to the Nonlinear Maxwell-Schrödinger System
References
On the Plate Equation with Exponentially Degenerating Stochastic Coefficients on the Torus
1 Introduction
2 Proof of Theorem 1
2.1 Introduction to Some Tools from Microlocal Analysis
2.2 Micro-Energy Estimates in Z1
2.3 Micro-Energy Estimates in Z2
2.4 Micro-Energy Estimates in Z3
2.5 Conclusion
3 Proof of Theorem 2
3.1 Step 1: Introduction of Some Auxiliary Functions and Series
3.2 Step 2: Construction of a Sequence of Oscillating Coefficients
3.3 Step 3: Construction of Auxiliary Functions
3.4 Step 4: Existence of ν-Loss of Regularity
References
Existence Results for Critical Problems Involving p-Sub-Laplacians on Carnot Groups
1 Introduction
2 The Functional Setting
3 The Mountain Pass Case
4 The Case with Linking Geometry
4.1 Statement of the Results
4.2 Proof of the Results
References
The Wodzicki Residue for Pseudo-Differential Operators on Compact Lie Groups
1 Introduction
2 Preliminaries
2.1 Pseudo-Differential Operators via Localisations
2.2 The Global Symbol in the Ruzhansky–Turunen Formalism
2.3 The Weak 1 Space L(1,∞)(G"0362G)
3 Proof of Theorem 1
References
New Characterizations of Harmonic Hardy Spaces
1 Introduction
2 A Description of the Harmonic Hardy Space h1 in D
3 A Description of the Harmonic Hardy Space H1 in G+
References
On the Solvability of the Synthesis Problem for Optimal Control Systems with Distributed Parameters
1 Introduction
2 Formulation of Synthesis Problem
3 Generalized Solution of Boundary Value Problem
4 On Solvability of the Synthesis Problem
5 Conclusion
References
On the Determination of a Coefficient of an Elliptic Equation via Partial Boundary Measurement
1 Introduction
2 Inverse Spectral Problem
3 Reconstruction Procedure
Appendix: Representation of Solution to Schrödinger Equation
References
Reconstruction from Boundary Measurements: ComplexConductivities
1 Introduction
2 Uniqueness of Schrödinger Inverse Problem
3 Preliminaries for Reconstruction
4 Boundary Integral Equation
5 From t to γ
6 Reconstruction of q from the Boundary Measurements γ
References