FY2025 Annual Report

Abdulla Unit Members 2025

Analysis and Partial Differential Equations
Professor Ugur Abdulla

Abstract

The Analysis and PDE unit aims 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 and Interdisciplinary Sciences include cancer detection through Electrical Impedance Tomography and optimal control theory; quantum optimal control theory and quantum technologies. 

1. Staff

  • Prof. Dr. Ugur Abdulla, Group Leader
  • Dr. Daniel Tietz, Researcher
  • Dr. Jose Rodrigues, Researcher
  • Dr. Denis Brazke, Researcher
  • Dr. Ian Miller, Researcher
  • Mr. Chenming Zhen, PhD Student

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 a 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 a new Wiener-type criterion for the full characterization 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 an asymptotic law for the conditional Brownian motion near the singularity point. The results are published in the following preprints: 

U. G. Abdulla, Wiener-type criterion for the removability of the fundamental singularity for the heat equation and its consequences, Math Arxiv:2501.00920v2, 2025; under review in Advances in Mathematics. 

U. G. Abdulla, Kolmogorov Problem and Wiener-type Criteria in Potential Theory, 2026; under review in Memoirs of the American Mathematical Society.

 

2.2 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 pursue a computational analysis of the biomedical problem of identifying a cancerous tumor at an early stage of development, based on EIT and optimal control of elliptic PDEs. Relying on the fact that the electrical conductivity of the cancerous tumor is significantly higher than that of healthy tissue, we consider an inverse EIT problem for identifying the conductivity map in the complete electrode model, based on the m current-to-voltage measurements on the boundary electrodes. A variational formulation as a PDE-constrained optimal control problem is introduced based on the novel idea of increasing the size of the input data by adding "voltage-to-current" measurements through various permutations of the single "current-to-voltage" measurement. The idea of permutation preserves the size of the unknown parameters at the expense of an increase in the number of PDE constraints. We apply a gradient projection method (GPM) based on the Fr\'echet differentiability in Besov-Hilbert spaces. Numerical simulations of 2D and 3D model examples demonstrate the sharp increase in the resolution of the cancerous tumor by increasing the number of measurements from m to m2.

The results are published in the following paper:

U.G. Abdulla and J.H. Rodrigues, Cancer Detection via Electrical Impedance Tomography and Optimal Control of Elliptic PDEs, Mathematics in Engineering, 2026, 8(2): 308-339.

 

2.3 Quantum Optimal Control in Biochemical Processes

We solve the quantum optimal control problem to find an external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state. In this context, the quantum coherence refers to the phase-coherent superposition of singlet and triplet states in a pair of free radicals, allowing them to exist simultaneously in both states in biochemical reactions. It is demonstrated that the Pontryagin Maximum Principle - a fundamental mathematical principle for the optimality of complex dynamical systems - turns out to be a fundamental principle for quantum coherence. We introduce a model of a filtered Schrodinger system, and demonstrate that trading off between the original non-filtered model with a bang-bang optimal magnetic field and the filtered model with a continuous in-time optimal magnetic field preserves quantum coherence. We develop a novel iterative regularization method for the identification of the bang-bang optimal control, and an associated simple continuous in-time optimal magnetic field input. The method has some similarity with the gradient ascent method, with the difference being that instead of moving in the gradient direction in the full control space, it generates the movement in the manifold of bang-bang control vectors guided by the Pontryagin Maximum Principle. 

The results are published in the following paper: 

U.G. Abdulla, J.H. Rodrigues, J.-J. Slotine, Quantum Optimal Control for Coherent Spin Dynamics of Radical Pairs via Pontryagin Maximum Principle, Quantum 10, 2096 (2026).

 

2.4. Dynamical Systems and Ergodic Theory

We prove a conjecture on the second minimal odd periodic orbits with respect to Sharkovsky ordering for the continuous endomorphisms on the real line. The conjecture was posed in U.G. Abdulla et al., International Journal of Bifurcation and Chaos, 27(5), 2017, 1-24. A \(2k+1\)-periodic orbit \(\beta_1<\beta_2<\cdots<\beta_{2k+1}, (k\geq 3)\) is called second minimal  for the map \(f\), if \(2k-1\) is a minimal period of  \(f|_{[\beta_1,\beta_{2k+1}]}\)in the Sharkovski ordering. The full classification of second minimal orbits is presented in terms of cyclic permutations and a directed graph of transitions. It is proved that second minimal odd orbits either have a Stefan structure like minimal odd orbits or have one of the \(4k-3\) types, each characterized by unique cyclic permutation and directed graph of transitions with accuracy up to inverses. The new concept of second minimal orbits and its classification has an important application toward the understanding of the universal structure of the distribution of the periodic windows in the bifurcation diagram generated by the chaotic dynamics of nonlinear maps on the interval.

The result is published in the following paper:

U.G. Abdulla, N.H. Iqbal, M.U. Abdulla, R.U. Abdulla, Classification of the Second Minimal Orbits in the Sharkovski Ordering, Axioms, 2025, 14(3), 222;  

 

3. Publications

3.1 Journals


  1. U.G. Abdulla and J.H. Rodrigues, Cancer Detection via Electrical Impedance Tomography and Optimal Control of Elliptic PDEs, Mathematics in Engineering, 2026, 8(2): 308-339
  2. U.G. Abdulla, J.H. Rodrigues, J.-J. Slotine, Quantum Optimal Control for Coherent Spin Dynamics of Radical Pairs via Pontryagin Maximum Principle, Quantum 10, 2096 (2026).
  3. U. G. Abdulla, N. H. Iqbal, M. U. Abdulla, R. U. Abdulla, Classification of Second Minimal Orbits in the Sharkovski Ordering,  Axioms, 2025, 14(3), 222; 

3.2 Preprints

  1. U. G. Abdulla, Wiener-type criterion for the removability of the fundamental singularity for the heat equation and its consequences, Math Arxiv:2501.00920v2, 2025; under review in Advances in Mathematics. 

  2. U. G. Abdulla, Kolmogorov Problem and Wiener-type Criteria in Potential Theory, 2026; under review in Memoirs of the American Mathematical Society.

  3. U.G. Abdulla and D. Brazke, Wiener Criterion at \(\infty\) for a class of linear degenerate elliptic equations, Math Arxiv: 2510:15249, 2025. 

  4. U.G. Abdulla and D. Tietz, Criterion for the removability of the fundamental singularity for the second-order parabolic PDEs, 2026.

3.3 Books and other one-time publications

Nothing to report

4. Invited Lectures and Conference Presentations

4.1 Invited Colloquium Lectures

4.2 Invited Conference Presentations

4.3 Contributed Conference Presentations

5. Intellectual Property Rights and Other Specific Achievements

Nothing to report.

6. Meetings and Events

6.1 Analysis and Partial Differential Equations Seminar Series 

6.2 Recent Advances in Potential Theory and Partial Differential Equations 2025