Harmonic Analysis and Partial Differential Equations

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. Preface 6 Contents 7 List of Contributors 9 The Wave Resolvent for Compactly Supported Perturbations of Minkowski Space 11 1 Introduction 11 1.1 Motivation 11 1.2 Main Result and Sketch of Proof 13 1.3 Structure of Paper 14 2 Essential Self-Adjointness 14 2.1 Preliminaries on Self-Adjointness 14 2.2 Preliminaries on Microlocal Analysis 17 2.3 Proof of Local Regularity 18 2.4 Generalization to Static Spacetimes 20 3 Uniform Microlocal Estimates 22 3.1 Uniform Wavefront Set 22 3.2 Uniform Resolvent Estimate 24 References 9 Smoothing Effect and Strichartz Estimates for Some Time-Degenerate Schrödinger Equations 28 1 Introduction 28 2 Smoothing Effect and Local Well-Posedness for the Class Lα,c 30 2.1 The Class Lα 31 2.2 The Class Lα,c 36 3 Strichartz Estimates and Local Well-Posedness for Lb 46 Appendix 50 References 52 On the Cauchy Problem for the Nonlinear Wave Equation with Damping and Potential 54 1 Introduction 54 2 Formulation of the Problem and Results 56 3 Preliminaries 59 4 Proof of Theorem 1 62 5 Proof of Theorem 2 65 References 9 Local Well-Posedness for the Scale-Critical Semilinear Heat Equation with a Weighted Gradient Term 71 1 Introduction and Main Results 71 2 Proofs of Theorems 1.2 and 1.3 75 References 9 On the Rellich Type Inequality for Schrödinger Operators with Singular Potential 84 1 Introduction 84 2 Preliminary Observations 86 3 Proof of Theorem 1 92 References 9 Global Solutions to the Nonlinear Maxwell-Schrödinger System 97 References 9 On the Plate Equation with Exponentially Degenerating Stochastic Coefficients on the Torus 103 1 Introduction 103 2 Proof of Theorem 1 112 2.1 Introduction to Some Tools from Microlocal Analysis 113 2.2 Micro-Energy Estimates in Z1 115 2.3 Micro-Energy Estimates in Z2 119 2.4 Micro-Energy Estimates in Z3 120 2.5 Conclusion 129 3 Proof of Theorem 2 132 3.1 Step 1: Introduction of Some Auxiliary Functions and Series 132 3.2 Step 2: Construction of a Sequence of Oscillating Coefficients 134 3.3 Step 3: Construction of Auxiliary Functions 135 3.4 Step 4: Existence of ν-Loss of Regularity 136 References 9 Existence Results for Critical Problems Involving p-Sub-Laplacians on Carnot Groups 140 1 Introduction 140 2 The Functional Setting 143 3 The Mountain Pass Case 145 4 The Case with Linking Geometry 149 4.1 Statement of the Results 149 4.2 Proof of the Results 151 References 52 The Wodzicki Residue for Pseudo-Differential Operators on Compact Lie Groups 157 1 Introduction 157 2 Preliminaries 160 2.1 Pseudo-Differential Operators via Localisations 160 2.2 The Global Symbol in the Ruzhansky–Turunen Formalism 161 2.3 The Weak 1 Space L(1,∞)(G"0362G) 165 3 Proof of Theorem 1 165 References 9 New Characterizations of Harmonic Hardy Spaces 171 1 Introduction 171 2 A Description of the Harmonic Hardy Space h1 in D 172 3 A Description of the Harmonic Hardy Space H1 in G+ 176 References 9 On the Solvability of the Synthesis Problem for Optimal Control Systems with Distributed Parameters 186 1 Introduction 186 2 Formulation of Synthesis Problem 187 3 Generalized Solution of Boundary Value Problem 188 4 On Solvability of the Synthesis Problem 194 5 Conclusion 198 References 9 On the Determination of a Coefficient of an Elliptic Equation via Partial Boundary Measurement 199 1 Introduction 199 2 Inverse Spectral Problem 203 3 Reconstruction Procedure 205 Appendix: Representation of Solution to Schrödinger Equation 207 References 52 Reconstruction from Boundary Measurements: ComplexConductivities 211 1 Introduction 211 2 Uniqueness of Schrödinger Inverse Problem 214 3 Preliminaries for Reconstruction 220 4 Boundary Integral Equation 224 5 From t to γ 234 6 Reconstruction of q from the Boundary Measurements γ 237 References 9