# Matthias Liero

Analysis of PDEs, Modeling, Simulations

Weierstrass Institute

Research Group "Partial Differential Equations"

MohrenstraĂźe 39, 10117

Germany

I am a post-doc at the WeierstraĂź Institute for Applied Analysis and Stochastics, Berlin, in the research group Partial Differential Equations of Alexander Mielke. I am also a member of the Berlin Mathematical School (BMS) in the Postdoctoral Faculty.

## Research interests

My mathematical expertise is in the field of nonlinear partial differential equations. In particular, the rigorous derivation of new effective models in various problems in natural sciences using novel mathematical techniques is one of my major interests. In my diploma I gave a rigorous justification of an evolutionary elastoplastic plate model for continuum mechanics using the notion of Gamma-convergence. In a joint work with U. Stefanelli (Vienna), we extended the Weighted-Energy-Dissipation principle from parabolic to hyperbolic equations to make them accessible to variational methods.

### Semiconductor modeling

In the recent years, the mathematical modeling, analysis, and simulation of optoelectronic devices, such as solar cells and organic light-emitting diodes, has become an essential application for me. I work closely with my colleagues A. Glitzky, Th. Koprucki, J. Fuhrmann, and D.H. Doan from WIAS on organic devices.

In a joint work with the Dresden Integrated Center for Applied Physics and Photonic Materials, in particular with A. Fischer, R. Scholz and S. Reineke, we derived a novel PDE model, involving the \(p(x)\)-Laplacian with discontinuous \(p(x)\), to describe the current and heat flow in organic light-emitting diodes. This was the first model to correctly predict S-shaped current-voltage characteristics with regions of negative differential resistance as observed in measurements.

Moreover, with M. Sawatzki and H. Kleemann from IAPP, we investigate the behavior and new concepts of organic transistors.

### Gradient flows and unbalanced optimal transport

However, also the more abstract theory behind partial differential equations is in the focus of my current work. One particular highlight of a recent joint work with A. Mielke (WIAS & HUB) and G. SavarĂ© (Milano) was the derivation and characterization of the so-called Hellinger-Kantorovich distance, which can be seen as a generalization of the famous Wasserstein distance to arbitrary measures.

In general, my scientific work is guided by the aim to strengthen the cooperation between analysis and its applications by inventing and further developing the mathematical foundations and techniques to make them applicable for practical questions in other sciences.

I am contributing to the software projects pdelib, ddfermi, VoronoiFVM.jl, ExtendableGrids.jl.