Matthias Liero

Weierstrass Institute

Research Group "Partial Differential Equations"

Mohrenstraße 39, 10117

Germany

I am a post-doc at the Weierstraß 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. Savaré (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.

selected publications

Fine properties of geodesics and geodesic lambda-convexity for the Hellinger–Kantorovich distance

Matthias Liero, Alexander Mielke, and Giuseppe Savaré

Weierstrass Institute Berlin Preprint, No. 2956 2022

Abs URL

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic lambda-convexity with respect to the Hellinger–Kantorovich distance.