748

Advances in and Partial Differential Equations

AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA

Donatella Danielli Irina Mitrea Editors Advances in Harmonic Analysis and Partial Differential Equations

AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA

Donatella Danielli Irina Mitrea Editors

748

Advances in Harmonic Analysis and Partial Differential Equations

AMS Special Session Harmonic Analysis and Partial Differential Equations April 21–22, 2018 Northeastern University, Boston, MA

Donatella Danielli Irina Mitrea Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan

2010 Subject Classification. Primary 31A10, 33C10, 35G20, 35P20, 35S05, 39B72, 42B35, 46E30, 76D03, 78A05.

Library of Congress Cataloging-in-Publication Data Names: AMS Special Session on Harmonic Analysis and Partial Differential Equations (2018 : Northeastern University). | Danielli, Donatella, 1966- editor. | Mitrea, Irina, editor. Title: Advances in harmonic analysis and partial differential equations : AMS special session on Harmonic Analysis and Partial Differential Equations, April 21-22, 2018, Northeastern University, Boston, MA / Donatella Danielli, Irina Mitrea, editors. Description: Providence, Rhode Island : American Mathematical Society, [2020] | : Con- temporary mathematics, 0271-4132 ; volume 748 | Includes bibliographical references. Identifiers: LCCN 2019040080 | ISBN 9781470448967 (paperback) | ISBN 9781470455163 (ebook) Subjects: LCSH: Harmonic analysis–Congresses. | Differential equations, Partial–Congresses. | AMS: – Two-dimensional theory – representations, integral opera- tors, integral equations methods. | . | Partial differential equations – General higher-order equations and systems – Nonlinear higher-order equations. | Partial differential equations – Spectral theory and eigenvalue problems – Asymptotic distribution of eigenvalues and eigenfunctions. | Partial differential equations – Pseudodifferential operators and other generalizations of partial differential operators – Pseudodifferential operators. | Difference and functional equations – Functional equations and inequalities – Systems of functional equa- tions and inequalities. | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – spaces arising in harmonic analysis. | – Linear func- tion spaces and their duals. | Fluid mechanics – Incompressible viscous fluids – Existence, uniqueness, and regularity theory. | Optics, electromagnetic theory – General – Geometric optics. Classification: LCC QA403 .R425 2018 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2019040080 DOI: https://doi.org/10.1090/conm/748/15051

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Preface vii BMO on shapes and sharp constants Galia Dafni and Ryan Gibara 1 Applications of harmonic analysis techniques to regularity problems of dissipative equations Mimi Dai and Han Liu 35

Two classical properties of the Bessel quotient Iν+1/Iν and their implications in pde’s Nicola Garofalo 57 On existence of dichromatic single element lenses Cristian E. Gutierrez´ and Ahmad Sabra 99 Free boundary regularity near the fixed boundary for the fully nonlinear obstacle problem Emanuel Indrei 147 The Poisson integral formula for variable-coefficient elliptic systems in rough domains Dorina Mitrea, Irina Mitrea, and Marius Mitrea 157 Variations on quantum ergodic theorems, II Michael Taylor 177

v

Preface

Problems arising in Partial Differential Equations are often models of a real- ity that is highly intricate from both analytic and geometric points of view. The tools required to treat these models need to be very sophisticated themselves and situated at the crossroads of several major fields of mathematics, including Har- monic Analysis (singular integral operators, Calder´on-Zygmund theory), Opera- tor Theory (spectral theory, functional ), /Several Com- plex Variables (CR , Cauchy-type integral operators, conformal and quasi- conformal mappings), (fast , ), Sci- entific Computing (interval analysis, ), and Geometric Theory (classes of sets of locally finite perimeter, quantitative versions of rectifi- ability, etc.). Combining techniques originating in these fields has proved to be extremely potent when dealing with a host of difficult and important problems in analysis. Indeed, there are many notable achievements in this direction whose degree of technical sophistication is truly breathtaking. The current volume focuses on new developments at the interface between Real and Complex Analysis, Harmonic Analysis, and Partial Differential Equations. It contains papers contributed by speakers in the Special Session on Harmonic Anal- ysis and Partial Differential Equations at the American Mathematical Society Sec- tional Meeting at Northeastern University, Boston, MA, April 21–22, 2018. The editors believe that it is imperative to raise the level of awareness of junior mathe- maticians, including graduate students and post-doctoral fellows, about the neces- sity of having a solid background in all of these disciplines, and the current volume offers a glimpse into a variety of current topics and research problems in these areas. Specifically,

• The paper BMO on shapes and sharp constants, by G. Dafni and R. Gibara, introduces and establishes fundamental properties and classical inequalities (including the John-Nirenberg inequality) of a new type of BMO space depending on an integrability parameter and a basis of shapes in the Euclidean setting. • The paper Applications of harmonic analysis techniques to regularity prob- lems of dissipative equations, by M. Dai and H. Liu, is a survey on the state of the art of Partial Differential Equations in fluid mechanics, with emphasis on establishing regularity results through conditions assumed on low frequency components of the solutions. • The paper Two classical properties of the Bessel quotient and their impli- cations in pde’s, by N. Garofalo, lies at the intersection between several different topics from Partial Differential Equations, Probability, Differ- ential Geometry, and Special Functions. Here the author uses classical

vii viii PREFACE

properties of the Bessel quotient to establish new and sharp results for a class of degenerate partial differential equations of parabolic type in the upper half-space, which arise in connection with the analysis of fractional heat operators. • The paper On existence of dichromatic single element lenses,byC.E. Guti´errez and A. Sabra, deals with the issue of existence of solutions for a modeling problem in geometric optics. In practical terms, this amounts to finding a refracting lens that is capable of reshaping a beam of dichromatic light from a source in a prescribed manner. • The paper Free boundary regularity near the fixed boundary for the fully nonlinear obstacle problem, by E. Indrei, is concerned with regularity prop- erties of the free boundary for obstacle type problems, a line of investiga- tion that has generated new developments in recent years. This is done through a blow-up classification procedure, which is of interest in its own. • The paper The Poisson integral formula for variable-coefficient elliptic systems in rough domains, by D. Mitrea, I. Mitrea, and M. Mitrea, es- tablishes Poisson integral formulas for solutions of second-order, homo- geneous, divergence-form elliptic systems with complex-valued Lipschitz coefficients for a very general class of non-smooth domains, which is sharp from the geometric measure theoretic point of view. • The paper Variations on quantum ergodic theorems, II,byM.Taylor, presents new quantum ergodic results for first-order positive, self-adjoint, elliptic pseudodifferential operators on compact Riemannian . Specifically, the work deals with scenarios in which the Hamiltonian flow generated by the symbol of the pseudodifferential operator in question is not necessarily ergodic. We hope you will enjoy this volume! Donatella Danielli Irina Mitrea Selected Published Titles in This Series

748 Donatella Danielli and Irina Mitrea, Editors, Advances in Harmonic Analysis and Partial Differential Equations, 2020 746 Erica Flapan and Helen Wong, Editors, and Geometry of Biopolymers, 2020 745 Federico Binda, Marc Levine, Manh Toan Nguyen, and Oliver R¨ondigs, Editors, Motivic Homotopy Theory and Refined Enumerative Geometry, 2020 744 Pieter Moree, Anke Pohl, L’ubom´ır Snoha, and Tom Ward, Editors, Dynamics: Topology and Numbers, 2020 743 H. Garth Dales, Dmitry Khavinson, and Javad Mashreghi, Editors, Complex Analysis and Spectral Theory, 2020 742 Francisco-Jes´us Castro-Jim´enez, David Bradley Massey, Bernard Teissier, and Meral Tosun, Editors, A Panorama of Singularities, 2020 741 Houssam Abdul-Rahman, Robert Sims, and Amanda Young, Editors, Analytic Trends in , 2020 740 Alina Bucur and David Zureick-Brown, Editors, Analytic Methods in Geometry, 2019 739 Yaiza Canzani, Linan Chen, and Dmitry Jakobson, Editors, Probabilistic Methods in Geometry, Topology and Spectral Theory, 2019 738 Shrikrishna G. Dani, Surender K. Jain, Jugal K. Verma, and Meenakshi P. Wasadikar, Editors, Contributions in and , 2019 737 Fernanda Botelho, Editor, Recent Trends in Operator Theory and Applications, 2019 736 Jane Hawkins, Rachel L. Rossetti, and Jim Wiseman, Editors, Dynamical Systems and Random Processes, 2019 735 Yanir A. Rubinstein and Bernard Shiffman, Editors, Advances in Complex Geometry, 2019 734 Peter Kuchment and Evgeny Semenov, Editors, Differential Equations, Mathematical Physics, and Applications, 2019 733 Peter Kuchment and Evgeny Semenov, Editors, Functional Analysis and Geometry, 2019 732 Samuele Anni, Jay Jorgenson, Lejla Smajlovi´c, and Lynne Walling, Editors, Automorphic Forms and Related Topics, 2019 731 Robert G. Niemeyer, Erin P. J. Pearse, John A. Rock, and Tony Samuel, Editors, Horizons of Fractal Geometry and Complex , 2019 730 Alberto Facchini, Lorna Gregory, Sonia L’Innocente, and Marcus Tressl, Editors, of Modules, and Categories, 2019 729 Daniel G. Davis, Hans-Werner Henn, J. F. Jardine, Mark W. Johnson, and Charles Rezk, Editors, Homotopy Theory: Tools and Applications, 2019 728 Nicol´as Andruskiewitsch and Dmitri Nikshych, Editors, Tensor Categories and Hopf Algebras, 2019 727 Andr´e Leroy, Christian Lomp, Sergio L´opez-Permouth, and Fr´ed´erique Oggier, Editors, Rings, Modules and Codes, 2019 726 Eugene Plotkin, Editor, Groups, Algebras and Identities, 2019 725 Shijun Zheng, Marius Beceanu, Jerry Bona, Geng Chen, Tuoc Van Phan, and Avy Soffer, Editors, Nonlinear Dispersive and Fluids, 2019 724 Lubjana Beshaj and Tony Shaska, Editors, Algebraic Curves and Their Applications, 2019 723 Donatella Danielli, Arshak Petrosyan, and Camelia A. Pop, Editors, New Developments in the Analysis of Nonlocal Operators, 2019

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/. CONM 748 amncAayi n ata ifrnilEquations Differential Partial and Analysis Harmonic

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 dif- ferential 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 , and the emphasis is on a host of new concepts, methods, and results. • ailiadMte,Editors Mitrea, and Danielli

ISBN 978-1-4704-4896-7

9 781470 448967 CONM/748