FY2020 Annual Report

Nonlinear Analysis Unit
Associate Professor Daniel Spector

 

Abstract

The (ongoing) pandemic demanded that most interactions in the Nonlinear Analysis Unit in FY20 be virtual.  In response, in place of the plan to have frequent visitors to the Unit, we launched the Fall 2020 and Spring 2021 Nonlinear Analysis Seminar Series.  These weekly lectures provided a substitute for usual seminar interactions and have been recorded in a Video Lecture Library that may be useful for many people in the future.  Alongside these seminars, the Unit members continued to work individually and in collaboration with researchers from around the world, with solid productivity.  It will be good when we can begin to work in person again, as mathematics suffers some from the lack of interaction, though in the meantime it is often our sanity to bear through the difficulties of life in the world today.  

1. Staff

  • Nikolas Chatzikonstantinou, Postdoctoral Scholar
  • Yu-wei Chen, Postdoctoral Scholar
  • Chiyo Eto, Administrative Assistant

2. Collaborations

2.1 Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers

  • Type of collaboration: Joint research
  • Researchers:
    • Professor  Scott J. Spector, Southern Illinois University

2.2 On the dimensional weak-type (1,1) bound for Riesz transforms

  • Type of collaboration: Joint research
  • Researchers:
    • Dr. Cody B. Stockdale, Clemson University

2.3 On Korn-Maxwell-Sobolev Inequalities

  • Type of collaboration: Joint research
  • Researchers:
    • Dr.  Franz Gmeineder, University of Bonn

2.4 Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers

  • Type of collaboration: Joint research
  • Researchers:
    • Professor  Scott J. Spector, Southern Illinois University

2.5 BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension

  • Type of collaboration: Joint research
  • Researchers:
    • Professor  Scott J. Spector, Southern Illinois University

2.6 An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

  • Type of collaboration: Joint research
  • Researchers:
    • Dr. Ángel D. Martínez, University of Toronto

2.7 Some remarks on \(L^1\) embeddings in the subelliptic setting

  • Type of collaboration: Joint research
  • Researchers:
    • Professor Steven G. Krantz, Washington University in St.Louis
    • Professor Marco Peloso, University of Milan

2.8 A boxing inequality for the fractional perimeter

  • Type of collaboration: Joint research
  • Researchers:
    • Professor Augusto C. Ponce, Université catholique de Louvain

2.9 A decomposition by non-negative functions in the Sobolev space \(W^{k,1}\)

  • Type of collaboration: Joint research
  • Researchers:
    • Professor Augusto C. Ponce, Université catholique de Louvain

 

3. Activities and Findings

 

3.1  Sparse domination results for compact operators

3.2 A topological toolbox for Sobolev maps

3.3 A priori geodesic diameter bounds for solutions to a variety of Plateau problems

  • Date: October 15, 2020
  • Venue: Online Seminar
  • Speakers: Professor Ulrich Menne, National Taiwan Normal University and National Center for Theoretical Sciences

3.4 Hardy--Littlewood--Sobolev inequality for \(p=1\)

  • Date: October 22, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

3.5 Scale-invariant tangent-point energies for knots and fractional harmonic maps

3.6 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 1, Plan.

  • Date: November 4, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

3.7 Trace inequalities of Sobolev type and nonlinear Dirichlet problems

  • Date: November 5, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Takanobu Hara, Hokkaido University　

3.8 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 2, Proofs (i).

  • Date: November 11, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

3.9 Algebraic methods in sum-product estimates and their applications

3.10 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 2, Proofs (ii).

  • Date: November 18, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

3.11 \(\mathscr{A}\)-quasiconvexity and partial regularity　

  • Date: November 19, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Franz Gmeineder, University of Bonn

3.12 Some classes of solutions of quasilinear elliptic equations with sub-natural growth terms.

3.13 Higher-integrability estimates for systems of PDEs with a non-linear pointwise elliptic constraint

3.14 Commutators and Bounded Mean Oscillation

3.15 Old and new in Compensated Compactness theory

  • Date: December 17, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Bogdan Raita, Max Planck Institute for Mathematics in the Sciences

3.16 Characterizations of non-radiating sources in the elastic waves

  • Date: February 26, 2021
  • Venue: Online Seminar
  • Speakers: Professor Jenn-Nan Wang, National Taiwan University

3.17 Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

  • Date: March 4, 2021
  • Venue: Online Seminar
  • Speakers: Associate Professor Yutaka Terasawa, Nagoya University

3.18 Inverse problems for semilinear elliptic equations

  • Date: March 11, 2021
  • Venue: Online Seminar
  • Speakers: Assistant Professor Yi-Hsuan Lin, National Chiao Tung Universiy

3.19 Asymptotic stability of the gradient flow of nonlocal energies

  • Date: March 18, 2021
  • Venue: Online Seminar
  • Speakers: Professor Nicola Fusco, University of Napoli Federico II

3.20 New perspectives on Sobolev norms

  • Date: March 24, 2021
  • Venue: Online Seminar
  • Speakers: Professor Haim Brezis, National Academy of Sciences

4. Publications

4.1 Journals

  1. Taylor's Theorem for Functionals on BMO with Application to BMO Local Minimizers (with Scott J. Spector), to appear in Quarterly of Applied Mathematics.
  2. On the dimensional weak-type (1,1) bound for Riesz transforms (with Cody B. Stockdale), to appear in Communications in Contemporary Mathematics.
  3. On Korn-Maxwell-Sobolev Inequalities (with Franz Gmeineder), J. Math. Anal. Appl. 502 (2021), no. 1, 125226, DOI.
  4. BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension (with Scott J. Spector), J. Elasticity 143 (2021), no. 1, 85-109, DOI.
  5. An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces (with Ángel D. Martínez), Adv. Nonlinear Anal. 10 (2021), no. 1, 877–894, DOI.
  6. Some remarks on \(L^1\) embeddings in the subelliptic setting (with Steven G. Krantz and Marco Peloso), Nonlinear Anal. 202 (2021), 112149, DOI.
  7. An Optimal Sobolev Embedding for \(L^1\). J. Funct. Anal. 279 (2020), no. 3, 108559, 26 pp, DOI.
  8. A boxing inequality for the fractional perimeter (with Augusto C. Ponce). Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020), no. 1, 107–141, DOI.
  9. A decomposition by non-negative functions in the Sobolev space \(W^{k,1}\) (with Augusto C. Ponce). Indiana Univ. Math. J. 69 (2020), no. 1, 151–169, DOI.
  10. New directions in harmonic analysis on \(L^1\). Nonlinear Anal. 192 (2020), 111685, 20 pp., DOI.

4.2 Books and other one-time publications

Nothing to report

4.3 Oral and Poster Presentations

([NOTE] *Seminars and workshops by OIST faculty/unit members (either with or without other speakers), either at OIST or at other institutions than OIST, should be included in the 4.3 Oral and Poster Presentations.

  1. Spector, D., BMO and Elasticity, Online Seminar 2010, Okinawa, Japan, July 3rd, (2020).
  2. Spector, D., Playing withElectricity and Magnetism, Online Seminar 2010, Okinawa, Japan, Aug 5th, (2020).
  3. Spector, D., Asia-Pacific Analysis andPDE Seminar, Online Seminar 2010, Okinawa, Japan, Oct 19th, (2020).

5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

6. Meetings and Events

6.1  Sparse domination results for compact operators

6.2 A topological toolbox for Sobolev maps

6.3 A priori geodesic diameter bounds for solutions to a variety of Plateau problems

  • Date: October 15, 2020
  • Venue: Online Seminar
  • Speakers: Professor Ulrich Menne, National Taiwan Normal University and National Center for Theoretical Sciences

6.4 Hardy--Littlewood--Sobolev inequality for \(p=1\)

  • Date: October 22, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

6.5 Scale-invariant tangent-point energies for knots and fractional harmonic maps

6.6 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 1, Plan.

  • Date: November 4, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

6.7 Trace inequalities of Sobolev type and nonlinear Dirichlet problems

  • Date: November 5, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Takanobu Hara, Hokkaido University　

6.8 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 2, Proofs (i).

  • Date: November 11, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

6.9 Algebraic methods in sum-product estimates and their applications

6.10 Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 2, Proofs (ii).

  • Date: November 18, 2020
  • Venue: Online Seminar
  • Speakers: Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

6.11 \(\mathscr{A}\)-quasiconvexity and partial regularity　

  • Date: November 19, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Franz Gmeineder, University of Bonn

6.12 Some classes of solutions of quasilinear elliptic equations with sub-natural growth terms.

6.13 Higher-integrability estimates for systems of PDEs with a non-linear pointwise elliptic constraint

6.14 Commutators and Bounded Mean Oscillation

6.15 Old and new in Compensated Compactness theory

  • Date: December 17, 2020
  • Venue: Online Seminar
  • Speakers: Dr. Bogdan Raita, Max Planck Institute for Mathematics in the Sciences

6.16 Characterizations of non-radiating sources in the elastic waves

  • Date: February 26, 2021
  • Venue: Online Seminar
  • Speakers: Professor Jenn-Nan Wang, National Taiwan University

6.17 Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

  • Date: March 4, 2021
  • Venue: Online Seminar
  • Speakers: Associate Professor Yutaka Terasawa, Nagoya University

6.18 Inverse problems for semilinear elliptic equations

  • Date: March 11, 2021
  • Venue: Online Seminar
  • Speakers: Assistant Professor Yi-Hsuan Lin, National Chiao Tung Universiy

6.19 Asymptotic stability of the gradient flow of nonlocal energies

  • Date: March 18, 2021
  • Venue: Online Seminar
  • Speakers: Professor Nicola Fusco, University of Napoli Federico II

6.20 New perspectives on Sobolev norms

  • Date: March 24, 2021
  • Venue: Online Seminar
  • Speakers: Professor Haim Brezis, National Academy of Sciences

7. Other

Nothing to report.