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Advances in Harmonic Analysis and Partial Differential Equations 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 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 Mathematics 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] | Series: 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: Potential theory – Two-dimensional theory – Integral representations, integral opera- tors, integral equations methods. | Special functions. | 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 – Function spaces arising in harmonic analysis. | Functional 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 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2020 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 Contents 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 calculus), Complex Analysis/Several Com- plex Variables (CR geometry, Cauchy-type integral operators, conformal and quasi- conformal mappings), Numerical Analysis (fast Fourier transform, wavelets), Sci- entific Computing (interval analysis, validated numerics), and Geometric Measure 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 manifolds. Specifically, the work deals with scenarios in which the Hamiltonian flow generated by the
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