Large-time behavior of reaction–diffusion-type equations

During my PhD studies under the supervision of Klemens Fellner I focused on the large-time behavior of a class of reaction–diffusion equations which arise in the modeling of semiconductors. By using an appropriate entropy functional and by deriving an inequality between the entropy and the entropy production functional, we were able to prove exponential convergence to equilibrium.
One of the investigated models considers a two-level system for electrons and holes with an intermediate level for immobile electrons in so-called trapped states. These additional electron states inside the bandgap of the semiconductor are due to impurities within the crystal of the semiconductor and typically facilitate the excitation of an electron from the valence band to the conduction band. Another model takes into account that electrons and holes are electrically charged and, therefore, influence their motion in a self-consistent manner. This additional coupling is described by Poisson's equation for the electrostatic potential leading to an additional drift term for electrons and holes. Recently, we also showed exponential equilibration for the self-consistent version of the trapped-states model.

Renormalized solutions to temperature-dependent PDE systems.

The existence of global renormalized solutions to temperature-dependent reaction–diffusion equations has been established in a joint work with Julian Fischer, Katharina Hopf, and Alexander Mielke. We focused on a PDE system in Onsager form for the concentrations of the involved species and the internal energy. Employing the internal energy (and not the temperature) as the main thermodynamic variable, facilitates some modeling issues; for example, reactions typically influence the temperature but leave the internal energy unchanged. Moreover, realistic models also have to account for cross-diffusive behavior between the concentrations and the internal energy to describe, for example, the Soret and the Dufour effect.
Global weak solutions are shown to exist for reactions with limited growth, while global renormalized solutions are constructed in the case that the reactions are not subject to any growth conditions.

Variance reduction for effective energies of random atomic lattices

At IST Austria, I have been working with Julian Fischer on a variance reduction technique for effective energies of random atomic lattices. A frequently employed approach for calculating material properties of random alloys is the method of special quasirandom structures which aims at constructing a periodic lattice with finite periodicity cell—the representative volume—which resembles the original lattice as closely as possible concerning certain statistics. The desired material properties are then calculated by solving the Thomas–Fermi–von Weizsäcker (TFW) equations for the electron density and by subsequently evaluating the TFW energy functional on the representative volume.
In our work, we have proven that this strategy indeed leads to a variance reduction of the TFW energy as compared to calculating the electron density and the TFW energy on a randomly chosen finite volume. We thereby derived the first rigorous variance reduction result related to an application of this selection approach to a nonlinear PDE model.

Material design for photovoltaics within Kohn–Sham DFT

Together with Gero Friesecke I also investigated a novel optimal control problem in the context of density functional theory (DFT) with possible applications to photovoltaics. Given a nuclear charge distribution as our control, one can calculate—by solving the Kohn–Sham equations—the separation of the generated electron–hole pair after the system has been excited by absorbing a photon. Finding the optimal control then corresponds to designing a material which gives rise to a maximal separation of positive and negative charge after a light-induced excitation.
We proved the existence of an optimal nuclear charge density in a 3D setting, and we observed a significant transfer of charge in a 1D numerical experiment when the system is excited from the ground state to the first excited state.