Oleksandr Misiats
Virginia Commonwealth University, VA·Mathematics & Statistics
Crumples in a sheet of paper, wrinkles in a curtain, defects in a liquid crystal or a superconductor all look different, but mathematically they share an origin: each pattern is the arrangement that minimizes some energy. This project studies that idea rigorously, building energy-minimizing models for materials and developing the mathematical techniques needed to prove which patterns are actually optimal.
- Fields
Mathematics & Statistics
Materials Science
Physics
- Topics
calculus of variations
singular perturbation
energy minimization
- Grants
- 1 known grant, through Aug 2027
What they work on
- Build and analyze energy-minimizing models of martensitic phase transitions — the crystal rearrangements behind the shape memory effect — to explain bending-induced effects reported experimentally in shape memory alloys.
- Develop a new approach, including treatment of anisotropy (direction-dependent material behavior), to prove that experimentally observed microstructural patterns in alloys are genuine energy minimizers.
- Show that minimizing the Ginzburg–Landau functional of superconductivity under semi-stiff boundary conditions yields solutions with vortices, and pin down their precise location near the domain boundary by matching upper and lower energy bounds.
- Extend classical homogenization theory to optimal control problems for the porous medium equation with stochastic forcing, where random disturbances perturb flow through a finely structured medium.
More about the research
The problems are "singularly perturbed" variational problems, meaning the energy contains a very small parameter whose presence forces fine-scale structure — stripes, layers, vortices — rather than a smooth solution. The aim is to explain experimentally observed structures and to supply tools that transfer between different materials settings.
Why it matters
Shape memory alloys, superconductors and porous materials are designed largely from experiment and intuition. Proving which microstructures minimize energy explains why the observed patterns form and where defects appear, which could inform the design of new materials such as shape memory alloys. The mathematical tools developed, including homogenization for randomly forced control problems, apply well beyond these specific systems.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Hung Tran
University of Wisconsin-Madison, WI·Mathematics & Statistics
Establish regularity and large-time behavior results for Hamilton-Jacobi equations, a class of nonlinear PDE from optimal control, and use them to prove that mass-conserving solutions of critical coagulation-fragmentation equations exist.
Paul Carter
University of California-Irvine, CA·Mathematics & Statistics·Early-career
Analyze when bistable planar interfaces, boundaries between two stable states, become unstable and form patterns in multi-component reaction-diffusion systems, motivated by desertification fronts in drylands.
Wilhelm Schlag
Yale University, CT·Mathematics & Statistics
Determine whether the kink soliton of the phi-4 model is asymptotically stable, extending a long-range scattering theory for potentials that have a threshold resonance.
The school
Virginia Commonwealth University: 19 labs on PI Pages, $524M a year on research, 438 research doctorates a year.
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