Analysis and Partial Differential Equations
Professor Ugur Abdulla
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; optimal control of reactive oxygen species in quantum biology.
1. Staff
- Prof. Dr. Ugur Abdulla, Group Leader
- Dr. Daniel Tietz, Researcher
- Dr. Jose Rodrigues, Researcher
- Dr. Prashant Goyal, Researcher
- Dr. Firoj Sk, 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\(\infty\).
Our recent research is focused on the characterization of the fundamental singularity of the heat equation. For \(h(x,t)\) being a fundamental solution of the heat equation in \(\mathbb{R}^{N+1}\) with the pole at the space-time origin \(O\), we introduce the notion of \(h\)-regularity (or \(h\)-irregularity) of the boundary point \(O\) of the arbitrary open subset of the upper half-space \(\mathbb{R}^{N+1}_+\) concerning heat equation, according as whether the \(h\)-parabolic measure of \(O\) is null (or positive). A necessary and sufficient condition for the removability of the fundamental singularity at \(O\), that is to say for the existence of a unique solution to the parabolic Dirichlet problem in a class \(O(h)\) is established in terms of the Kolmogorov test for the \(h\)-regularity of \(O\). From a topological point of view, the test presents the parabolic minimal thinness criterion of sets near \(O\) in parabolic minimal fine topology. Precisely, the open set is a deleted neighborhood of \(O\) in parabolic minimal fine topology if and only if \(O\) is \(h\)-irregular. From the probabilistic point of view, the test presents asymptotic law for the \(h\)-Brownian motion near \(O\). The following paper is in stage of submission:
U. G. Abdulla, Criteria for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences, to be submitted
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 results are to be 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 vis 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 Optimal Control in Quantum Biology
The overarching goal of this project is to develop and exploit the advanced methods of quantum optimal control theory to reveal the deep relationship between functional optimization of internal hyperfine parameters in flavoproteins and/or external magnetic field intensity input to maximize the quantum singlet yield in biochemical processes. The aim is to unveil the groundbreaking role of quantum coherence in biochemical processes. In the following recent paper, the coherent spin dynamics of radical pairs in biochemical reactions modeled by the Schrodinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. We considered the problem of identification of the constant magnetic field and internal hyperfine parameters which optimize the quantum singlet-triplet yield of the radical pair system. We developed qlopt algorithm to identify optimal values of a 3-dimensional external electromagnetic field vector and 3- or 6-dimensional hyperfine parameter vector which optimize the quantum singlet-triplet yield for the spin dynamics of radical pairs in 8- or 16-dimensional Schrodinger system corresponding to one- and two-proton cases respectively. Results demonstrate that the quantum singlet-triplet yield of the radical pair system can be significantly reduced if optimization is pursued simultaneously for the external magnetic field and internal hyperfine parameters. The results represent a crucial step to affirm the direct connection between hyperfine optimization and quantum coherence.
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.
opens a great new perspective for the breakthrough in understanding non-isolated singularities performed by solutions of second-order elliptic and parabolic PDEs at the finite boundary points and the point at
3. Publications
3.1 Journals
- 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
- 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
- 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/
- 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.
3.2 Books and other one-time publications
Nothing to report
4. Invited Lectures and Conference Presentations
4.1 Invited Colloquium Lectures
- U.G. Abdulla, The Wiener-type Criterion at \(\infty\) for the Elliptic and Parabolic PDEs and its Consequences, Department of Mathematics Colloquium, University of Central Florida, 3-4 pm, Friday, September 23, 2022, Orlando, Florida, USA.
- U.G. Abdulla, Potential Theory, Kolmogorov Problem and the Legacy of Wiener, OIST lunch time seminar, 12-1 pm, Wednesday, October 19, 2022.
- U.G. Abdulla, Optimal control of magnetic and hyperfine parameters to maximize quantum yield in radical pair reactions: a Quantum Biology approach, Johns Hopkins University Applied Physics Lab, Seminar, 11 am -12, Monday, November 14, 2022, Lauren, Maryland, USA.
- U.G. Abdulla, Classification of Singularities for the Elliptic and Parabolic PDEs and its Measure-theoretical Topological and Probabilistic Consequences, Department of Mathematics Colloquium, University of Virginia, 4-5 pm, Thursday, February 2, 2023, Charlottesville, Virginia, USA.
- U.G. Abdulla, Classification of Singularities for the Elliptic and Parabolic PDEs and its Measure-theoretical, Topological and Probabilistic Consequences, Department of Mathematics, University of Memphis, Colloquium, 4-5 pm, Friday, March 17, 2023, Memphis, Tennessee, USA.
4.2 Invited Conference Presentations
- U.G. Abdulla, Bang-bang Optimal Control in Spin Dynamics of Radical Pairs in Quantum Biology, 10th International Conference Inverse Problems: Modeling & Simulation, May 22-28, 2022, Malta
- U.G. Abdulla, On the Wiener Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences, JMM 2023, Joint Mathematics Meeting, January 4, 2023, Boston, Massachusetts, USA.
- 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 Mathematics in the Sciences (MiS) seminar series
- Date: January 18, 2023
- Venue: OIST Campus L4E48, 12-1 pm
- Speaker: Professor Ugur Abdulla (OIST)
- Title: Optimal Control of Magnetic and Hyperfine Parameters to Maximize Quantum Yield in Radical Pair Reactions: a Quantum Biology Approach
- Date: January 25, 2023
- Venue: OIST Campus L4E48, 12-1 pm
- Speaker: Ms. Friederike Metz (OIST, Busch unit, Ph.D. student)
- Title: Self-correcting Quantum Many-body Control using Reinforcement Learning with Tensor Network
- Date: March 3, 2023
- Venue: OIST Campus L4E48, 12-1 pm
- Speaker: Dr. Feng Li (Uppsala University, Sweden)
- Title: Wiener-type Criterion for the Boundary Holder Regularity for the Fractional Laplacian
- Date: March 8, 2023
- Venue: OIST Campus L4E48, 12-1 pm
- Speaker: Professor Eliot Fried (OIST)
- Title: Mobius Bands Obtained by Isometrically Deforming Circular Helicoids
- Date: March 9, 2023
- Venue: OIST Campus L4E48, 12-1 pm
- Speaker: Dr. Dingqun Deng (Beijing Institute of Mathematical Sciences, China)
- Title: Kinetic Theory: Stability, Regularity, and Spectral Analysis of the Boltzmann Equation.
- 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
7. Other
Nothing to report.
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