Advances in Harmonic Analysis and Partial Differential Equations

✍ Scribed by Donatella Danielli (editor), Irina Mitrea (editor)

• Publisher: American Mathematical Society
• Year: 2020
• Tongue: English
• Leaves: 212
• Series: Contemporary Mathematics
• Category: Library

✦ Synopsis

This volume contains the proceedings of the AMS Special Session on Harmonic Analysis and Partial Differential Equations, held from April 21-22, 2018, at Northeastern University, Boston, Massachusetts. The book features a series of recent developments at the interface between harmonic analysis and partial differential equations and is aimed toward the theoretical and applied communities of researchers working in real, complex, and harmonic analysis, partial differential equations, and their applications. The topics covered belong to the general areas of the theory of function spaces, partial differential equations of elliptic, parabolic, and dissipative types, geometric optics, free boundary problems, and ergodic theory, and the emphasis is on a host of new concepts, methods, and results.

✦ Table of Contents

Cover
Title page
Contents
Preface
𝐁𝐌𝐎 on shapes and sharp constants
1. Introduction
2. Preliminaries
3. 𝐵𝑀𝑂 spaces with respect to shapes
4. Shapewise inequalities on 𝐵𝑀𝑂
5. Rearrangements and the absolute value
6. Truncations
7. The John-Nirenberg inequality
8. Product decomposition
Acknowledgments
References
Applications of harmonic analysis techniques to regularity problems of dissipative equations
1. Overview
2. Harmonic analysis tools
3. Low modes regularity criteria for fluid equations
Acknowledgments
References
Two classical properties of the Bessel quotient 𝐼_{𝜈+1}/𝐼_{𝜈} and their implications in pde’s
1. Introduction
2. The Bessel semigroup
3. A curvature-dimension inequality
4. An inequality of Li-Yau type for the Bessel semigroup
5. A comparison with the results of Chiarenza-Serapioni and of Epstein-Mazzeo
6. A sharp Harnack inequality for the parabolic extension problem
7. Monotonicity formulas of Struwe and Almgren-Poon type for the Bessel semigroup
8. Appendix: The modified Bessel function 𝐼_{𝜈}(𝑧)
Acknowledgments
References
On the existence of dichromatic single element lenses
1. Introduction
2. Preliminaries
3. The collimated case: Problem A
3.1. Estimates of the upper surfaces for two colors
4. First order functional differential equations
4.1. Uniqueness of solutions
5. One point source case: Problem B
5.1. Two dimensional case, 𝑤∈Ω.
5.2. Derivation of a system of functional equations from the solvability of problem B in the plane.
5.3. Solutions of (5.4)yield local solutions to the optical problem.
5.4. On the solvability of the algebraic system (4.3)
5.5. Existence of local solutions to (5.4)
Acknowledgments
References
Free boundary regularity near the fixed boundary for the fully nonlinear obstacle problem
1. Introduction
2. Non-transversal intersection and classification of blow-up limits
3. 𝐶¹ regularity
4. Appendix
Acknowledgments
References
The Poisson integral formula for variable-coefficient elliptic systems in rough domains
1. Introduction
2. Preliminary matters
3. Proof of main result
Acknowledgment
References
Variations on quantum ergodic theorems, II
1. Introduction
2. Quantum ergodic theorems with discontinuous symbols
2.1. Quantization of discontinuous symbols
2.2. Weyl law
2.3. Quantum ergodic theorems
2.4. 𝐿^{𝑝} eigenfunction estimates and quantum ergodic theorems
3. Quantum ergodic theorems for conjugates 𝑒^{-𝑖𝑡Λ}𝐴𝑒^{𝑖𝑡Λ}
3.1. Complements to Weyl laws and Egorov’s theorem
3.2. Proof of Theorem 3.0.3
3.3. Comments and examples
References
Back Cover