FY2023 Annual Report

Analysis and Partial Differential Equations
Professor Ugur Abdulla

Four group members in front of a blackboard

Abstract

The aim of the Analysis and PDE unit is to reveal and analyze the mathematical principles reflecting natural phenomena expressed by partial differential equations. Research focuses on the fundamental analysis of PDEs, regularity theory of elliptic and parabolic PDEs, with special emphasis on the regularity of finite boundary points and the point at \(\infty\), its measure-theoretical, probabilistic, and topological characterization, well-posedness of PDE problems in domains with non-smooth and non-compact boundaries, global uniqueness, analysis and classification of singularities, asymptotic laws for diffusion processes, regularity theory of nonlinear degenerate and singular elliptic and parabolic PDEs, free boundary problems, optimal control of free boundary systems with distributed parameters. Some of the current research projects in Applied Mathematics include cancer detection through Electrical Impedance Tomography and optimal control theory; identification of parameters in large-scale models of systems biology; quantum optimal control of biochemical processes.

1. Staff

  • Prof. Dr. Ugur Abdulla, Group Leader
  • Dr. Daniel Tietz, Researcher
  • Dr. Jose Rodrigues, Researcher
  • Mr. Chenming Zhen, PhD Student
  • Ms. Miwako Tokuda Administrative Assistant

2. Activities and Findings

2.1 Potential Theory and PDEs

One of the major problems in the Analysis of PDEs is understanding the nature of singularities of solutions of PDEs reflecting the natural phenomena. Solution of the Kolmogorov Problem and new Wiener-type criterion for the regularity of \(\infty\) opened a great new perspective for the breakthrough in understanding non-isolated singularities performed by solutions of the elliptic and parabolic PDEs at the finite boundary points and the point at \(\infty\).

We solved an outstanding open problem and proved the necessary and sufficient condition for the removability of the fundamental singularity, and equivalently for the unique solvability of the singular Dirichlet problem for the heat equation. In the measure-theoretical context the criterion determines whether the parabolic measure of the singularity point is null or positive. From a topological point of view, the result presents the parabolic minimal thinness criterion of sets in parabolic minimal fine topology. From the probabilistic point of view, the test presents asymptotic law for the \(h\)-Brownian motion near the singularity point. The following preprint is in the review process:

U. G. Abdulla, Removability of the Fundamental Singularity for the Heat Equation and its Consequences. I. Kolmogorov-Petrovsky-type test, 2023, Math Arxiv:2312.06413.

 

2.2 Qualitative Theory and Regularity for Nonlinear PDEs

One of the key problems of the qualitative theory of degenerate and singular parabolic PDEs is understanding the smoothness and evolution properties of interfaces. The aim of the research project was to pursue a full classification of the short-time behavior of the solution and the interfaces in the Cauchy problem for the nonlinear singular parabolic PDE

\(u_t-\Delta u + bu^\beta=0, x\in \mathbb{R}^N, t > 0\)

with a nonnegative initial function \(u_0\) such that

\(supp \ u_0 = \{ |x|< R\}, u_0\sim C(R-|x|)^\alpha, as |x| \to R-0,\)

where \(0 < m < 1, b, \beta, C, \alpha > 0\) . Depending on the relative strength of the fast diffusion and absorption terms the problem may have infinite \((\beta \geq m)\) or finite \((\beta < m)\) speed of propagation. In the latter case, inerface surface \(t=\eta(x)\) may shrink, expand or remain stationary depending on the relative strength of the fast diffusion and absorption terms near the boundary of support, expressed in terms of parameters \(m, \beta, \alpha\) and \(C\). In all cases, we prove existence or non-existence of interfaces, explicit formula for the interface asymptotics, and local solution near the interface or at infinity. The results are published in a paper

U.G. Abdulla and A. Abuweden, Interface Development for the Nonlinear Degenerate Multidimensional Reaction-Diffusion Equations. II. Fast Diffusion versus Absorption, Nonlinear Differ. Equ. Appl. 30, 38 (2023). https://doi.org/10.1007/s00030-023-00847-x

 

2.3 Sobolev Spaces

The concept of Sobolev Spaces became a trailblazing idea in many fields of mathematics. The goal of this project is to gain insight into the embedding of the Sobolev spaces into Holder spaces - a very powerful concept that reveals the connection between weak differentiability and integrability (or weak regularity) of the function with its pointwise regularity. It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Holder spaces holds if the integrability exponent is higher than the space dimension. Otherwise speaking, one can trade one degree of weak regularity with an integrability exponent higher than the space dimension to upgrade the pointwise regularity to Holder continuity. In my recent research paper, the embedding of the Sobolev functions into the Holder spaces is expressed in terms of the weak differentiability requirements independent of the integrability exponent. Precisely, the question asked is what is the minimal weak regularity degree of Sobolev functions which upgrades the pointwise regularity to Holder continuity independent of the integrability exponent. The paper reveals that the anticipated subspace is a Sobolev space with dominating mixed smoothness, and proves the embedding of those spaces into Holder spaces. The new method of proof is based on the generalization of the Newton-Leibniz formula to \(n\)-dimensional rectangles:

\(u(x')-u(x)=\sum\limits_{k=1}^n\sum\limits_{\substack{i_1,...,i_k=1 \\ i_1 < ... < i_k}}^n \ \int\limits_ {P_{i_1\dots i_k}}\frac{\partial^k u(\eta)}{\partial x_{i_1} \cdots \partial x_{i_k}} \,d\eta_{i_1}\cdots\,d\eta_{i_k}\)

and inductive application of the Sobolev trace embedding results. Counterexamples demonstrate that in terms of the minimal weak regularity degree Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions which preserve the generalized Newton-Leibniz formula, and upgrade the pointwise regularity to Holder continuity. 

The result is published in

U.G. Abdulla, Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Holder Spaces, AIMS Mathematics, 2023, Volume 8, Issue 9: 20700-20717. http://www.aimspress.com/article/doi/10.3934/math.20231055 

 

2.4 Mathematical Biosciences: Cancer Detection via Electrical Impedance Tomography (EIT) and Optimal Control Theory

The goal of this project is to develop a new mathematical framework utilizing the theory of PDEs, inverse problems, and optimal control of systems with distributed parameters for a better understanding of the development of cancer in the human body, as well as the development of effective methods for the detection and control of tumor growth.

In a recent paper, we consider the Inverse EIT problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the \(m\) electrodes for a given electrode current. The variational formulation is introduced as a PDE-constrained coefficient optimal control problem in Sobolev spaces with dominating mixed smoothness. Electrical conductivity and boundary voltages are control parameters, and the cost functional is the \(L_2\)-norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. The existence of the optimal control and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.

The results are published in

U.G. Abdulla and S. Seif, Discretization and Convergence of the EIT Optimal Control Problem in Sobolev Spaces with Dominating Mixed Smoothness, Contemporary Mathematics, Volume 784, 2023 https://www.ams.org/books/conm/784/

 

2.5 Quantum Optimal Control in Biochemical Processes

The overarching goal of this project is to unveil the groundbreaking role of quantum coherence in biochemical processes. Optimal control of the external electromagnetic field input for the maximization of the quantum triplet-singlet yield of the radical pairs in biochemical reactions modeled by Schrodinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. Frechet differentiability and Pontryagin Maximum Principle in Hilbert space is proved and the bang-bang structure of the optimal control is established. A closed optimality system of nonlinear differential equations for the identification of the bang-bang optimal control is revealed. Numerical methods for the identification of the bang-bang optimal control based on the Pontryagin maximum principle are developed. Numerical simulations are pursued, and the convergence and stability of the numerical methods are demonstrated. The results contribute towards understanding the structure-function relationship of the putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices. The following preprint is in the review process. 

U.G. Abdulla, J. Rodrigues, R. Abdulla, P. Jimenez, Ch. Zhen and C. Martino, Bang-bang Optimal Control in Coherent Spin Dynamics of Radical Pairs in Quantum Biology, submitted. 

 

3. Publications

3.1 Journals

  1. U.G. Abdulla, Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Holder Spaces, AIMS Mathematics, 8, 9(2023), 20700-20717. http://www.aimspress.com/article/doi/10.3934/math.20231055 
  2. U.G. Abdulla and A. Abuweden, Interface Development for the Nonlinear Degenerate Multidimensional Reaction-Diffusion Equations. II. Fast Diffusion versus Absorption, Nonlinear Differ. Equ. Appl. 30, 38 (2023). https://doi.org/10.1007/s00030-023-00847-x
  3. U.G. Abdulla and S. Seif, Discretization and Convergence of the EIT Optimal Control Problem in Sobolev Spaces with Dominating Mixed Smoothness, Contemporary Mathematics, Volume 784, 2023 https://www.ams.org/books/conm/784/
  4. C.F. Martino, P. Jimenez, J. Goldfarb, U.G. Abdulla, Optimization of Parameters in Coherent Spin Dynamics of Radical Pairs in Quantum Biology, PLoS ONE 18(2), 2023. https://doi.org/10.1371/journal.pone.0273404.

3.1 Preprints

  1. U. G. Abdulla, Removability of the Fundamental Singularity for the Heat Equation and its Consequences. I. Kolmogorov-Petrovsky-type test, Math Arxiv:2312.06413.
  2. U.G. Abdulla, J. Rodrigues, R. Abdulla, P. Jimenez, Ch. Zhen and C. Martino, Bang-bang Optimal Control in Coherent Spin Dynamics of Radical Pairs in Quantum Biology, submitted.

3.2 Books and other one-time publications

Nothing to report

4. Invited Lectures and Conference Presentations

4.1 Invited Colloquium Lectures

  1. Department of Mathematics, University of Memphis, Colloquium, 4-5 pm, Friday, March 17, 2023, Memphis, Tennessee.
  2. Department of Mathematics, University of Pittsburgh, Analysis and PDE Seminar, 4-5 pm, Monday, October 23, Pittsburgh, Pennsylvania.
  3. Department of Mathematics, University of Alabama at Birmingham, Colloquium, 2:30-3:30 pm, Friday, November 17, Birmingham, Alabama.
  4. Department of Mathematics, University of Florida, Colloquium, 4-5 pm, Monday, November 20, Gainesville, Florida.
  5. Department of Mathematics, University of Chicago, Calderon-Zygmund Analysis Seminar, 4-5 pm, Thursday, March 28, 2024, Chicago, Illinois.
  6. Department of Mechanical Engineering, Massachusetts Institute of Technology, Seminar, Room 2-105, 4-5 pm, Wednesday, April 24, 2024.  

4.2 Invited Conference Presentations

  1. D. Tietz, U.G. Abdulla, Wiener Criterion at \(\infty\) for Divergence form Parabolic Operators with C1-Dini Continuous Coefficients, JMM 2024, Joint Mathematics Meeting, January 3-6, 2024, San-Francisco, USA.
  2. U.G. Abdulla, Cancer Detection via Electrical Impedance Tomography and PDE Constrained Optimal Control in Sobolev Spaces, Interdisciplinary Science Conference at Okinawa, ISCO 2023 - Physics and Mathematics meet Medical Science, 27 February - 3 March 2023, OIST, Okinawa, Japan.

5. Intellectual Property Rights and Other Specific Achievements

Nothing to report

6. Meetings and Events

6.1 Analysis and Partial Differential Equations Seminar Series 

  • Date: January 11, 2024
  • Venue: Zoom, 11:00 -11:50 
  • Speaker: Dr. Animesh Biswas (University of Nebraska-Lincoln)
  • Title: Symmetry of hypersurfaces with ordered nonlocal mean curvature
  • Date: January 12, 2024
  • Venue: Zoom, 9:00 -9:80 
  • Speaker: Dr. Joshua Flynn (McGill University)
  • Title: Sharp functional inequalities in geometry and PDEs
  • Date: January 12, 2024
  • Venue: Zoom, 10:00 -10:50 
  • Speaker: Dr.  Sean Patrick Douglas (University of Missouri)
  • Title: Kato-Ponce Inequality With Multiple (A_{\vec P}) Weights
  • Date: January 12, 2024
  • Venue: Zoom, 12:00 -12:50 
  • Speaker: Prof. Lei Zhang (University of Florida)
  • Title: Asymptotic behavior of solutions to the Yamabe equation in low dimensions,
  • Date: January 12, 2024
  • Venue: Zoom, 13:00 -13:50 
  • Speaker: Dr. Gongping Niu (University of California San Diego) 
  • Title: The existence of singular isoperimetric hypersurfaces
  • Date: January 12, 2024
  • Venue: Zoom, 15:00 -15:50 
  • Speaker:  Dr. Zetao Cheng (Tsinghua University)
  • Title: Asymptotic behavior of low energy nodal solutions of the Lane-Emden problem,
  • Date: January 12, 2024
  • Venue: Zoom, 16:00 -16:50 
  • Speaker: Dr. Fengrui Yang (University of Freiburg)
  • Title: Prescribed curvature measure problem in the hyperbolic space
  • Date: January 19, 2024
  • Venue: Zoom, 9:00 -10:00 
  • Speaker: Dr.  Kuan-Ting Yeh, University of Washington
  • Title: The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Summarization
  • Date: January 19, 2024
  • Venue: Zoom, 16:00 -17:00 
  • Speaker: Dr. Denis Brazke, Heidelberg University
  • Title:  Asymptotic analysis for a sharp interface model related to pattern formation in bio membranes
  • Date: February 22, 2024
  • Venue: Zoom, 8:00 -9:00 
  • Speaker: Mr. Le, Minh (Michigan State University)
  • Title: Chemotaxis systems under Nonlinear Neumann boundary conditions.
  • Date: January 22, 2024
  • Venue: Zoom, 13:00 -14:00
  • Speaker: Mr. Kyeong, Jeongsu (Temple University)
  • Title: The Neumann Problem for the bi-Laplacian in Infinite Sectors. 
  • Date: January 22, 2024
  • Venue: Zoom, 14:00 -15:00
  • Speaker: Mr. Kumar, Nitin (Indian Institute of Technology Delhi)
  • Title: The Wavelet collocation methods for fractional optimal control problem.

6.2 Mathematics in the Sciences (MiS) seminar series

  • Date: April 5, 2023
  • Venue: OIST Campus L4E48, 12-1 pm
  • Speaker: Professor Jonathan Woodward (University of Tokyo)
  • Title: Quantum Biology: Radical Pairs under the Microscope
  • Date: January 17, 2024
  • Venue: OIST Campus L4E48, 12:00-113:00
  • Speaker: Prof. Jean-Jacques Slotine (Massachusetts Institute of Technology)
  • Title: Stable adaptation and learning in dynamical systems
  • Date: April 10, 2024
  • Venue: OIST Campus L4E48, 12-1 pm
  • Speaker: Professor Ugur G. Abdulla (OIST)
  • Title: Cancer Detection via Electrical Impedance Tomography and Optimal Control Theory: invitation to interdisciplinary research5, 2023

6.3 Summer Graduate School

  • On 12-17 June 2023 Summer Graduate School is organized at OIST which provided six days of concentrated study of topics in Analysis and Partial Differential Equations. The school offered two minicourses and several plenary lectures on the topics at the forefront of current research in Partial Differential Equations and Potential Theory. 

6.4 Workshop: Recent Advances in Quantum Biology

  • Quantum mechanics is central to our understanding of both physics and chemistry, while in biology existing theories and models have largely treated biological systems as behaving according to the rules of classical physics (albeit made up of particles that obey the rules of quantum mechanics). However, in recent years, across a range of different biological phenomena, scientists have started to understand that such a simplification may be masking some of biology’s most remarkable abilities, recently leading to the rise of a new interdisciplinary research area known as quantum biology.

    Recently the field of quantum biology has been advanced by experimental and theoretical efforts leading to fascinating  discoveries suggesting that the hidden quantum effects may shape biological functions of living organisms, as epitomized by photosynthesis, enzyme calalysed reactions, and magnetic field effects on spin-dependent biochemical reactions.

    One day workshop will address some recent advances in the interdisciplinary field of quantum biology such as microspectroscopic detection of magnetic field sensitive radical pair processes in biological systems, reactive oxygen production in electron transfer flavoprotein, and development of mathematical foundation based on quantum optimal control theory in Hilbert-Sobolev spaces, theoretically proved optimality conditions implementing tools of infinite-dimensional calculus, Fréchet differentiability, Pontryagin Maximum Principle to comprehend coherent spin dynamics of radical pairs. 
  • Date: July 26, 2023
  • Venue: OIST Campus B250, 9:00 - 9:30 
  • Speaker: Professor Jonathan Woodward (University of Tokyo)
  • Title: What is Quantum Biology and why does it matter?
  • Date: July 26, 2023
  • Venue: OIST Campus B250, 11:00 - 11:45 
  • Speaker: Professor Carlos F. Martino (Johns Hopkins University Applied Physics Lab)
  • Title: The Quantum Biology of Reactive Oxygen Production in Electron Transfer Flavoprotein
  • Date: July 26, 2023
  • Venue: OIST Campus B250, 14:00 - 14:45 
  • Speaker: Professor Ugur G. Abdulla (Okinawa Institute of Science and Technology)
  • Title: Pontryagin Maximum Principle in optimal control of coherent spin dynamics of radical pairs in Quantum Biology I. Mathematical foundation and optimality conditions.
  • Date: July 26, 2023
  • Venue: OIST Campus B250, 15:15 - 16:00 
  • Speakers:
    Mr. Pablo Jimenez, Instituto Balseiro, Universidad Nacional de Cuyo, Phd Student
    Dr. Jose Rodriques, OIST, Analysis and Partial Differential Equations Unit(Abdulla Unit),Staff Scientist
    Mr. Chenming Zhen, OIST, Phd Student
  • Title:Pontryagin Maximum Principle in optimal control of coherent spin dynamics of radical pairs in Quantum Biology II. Computational Analysis

7. Other

Nothing to report.

File Attachments

IMG_0740.pdf