# Research of Carina Geldhauser

## My research deals with phenomena arising in physics and image processing. To solve the questions arising there, I use tools from applied analysis, partial differential equations, and probability theory. My goal is to capture the qualitative and quantitative behavior of solutions of nonlinear PDEs, which can be perturbed by noise.

.

Up to now, I have studied the existence and behavior of solutions to nonlinear PDEs in two different flavors:

### Applied Stochastic Analysis

Under this header I would like to put some recent ``excursions'' I did, on problems which are related to my main research areas. Just to mention on example: There is a large literature on optimization under PDE constraints, but no-one seems to have considered optimizing the fractional order of, say, a fractional heat equation. In a - maybe unconventional - setting from the Stochastic Analysis point of view, Enrico Valdinoci and me considered a optimal control problem with SPDE constraint with the fractional power as a control variable.

### Interacting Particle Systems

Interacting particle systems model complex phenomena in natural and social sciences. These phenomena involve a large number of interrelated components, which are modeled as particles confined to a lattice. I study so-called interacting diffusion models, i.e. I consider continuous on-site variables. Therefore my models take the form of a system of coupled stochastic differential equations. My goal is to describe the macroscopic behavior of the interacting diffusion as a nonlinear stochastic partial differential equation.

### Gradient flows of non-convex potentials

Gradient flows describe the evolution of a system as the steepest descent of an energy potential. This means that our system is minimizing its energy over time. Non-convex potentials, appearing for example in phase transitions or image processing, give rise to forward-backward parabolic PDEs. I try to determine the regime of initial data under which we can prove existence of solutions to such PDEs. Moreover, I study the behavior and properties of solutions to forward-backward parabolic PDEs.

#### Research updates

*On the right are some new related to my research, i.e. talks, new preprints etc. You can find my preprints on the servers of Calculus of Variations and Geometric Measure Theory at Pisa and arxiv.org*

For links to the publications, see the header Publications

### June 2017

Talk at conference Interacting Systems and SPDEs, Sheffield

### June 2017

Talk at research seminar Analysis at University of Augsburg

### March 2017

My latest work in collaboration with Enrico Valdinoci (Melbourne and Milano) is now on arxiv.org

### December 2016

Sometimes joy and regret are so close together: I just got the news that my grant application for a collaboration between WIAS Berlin and Chebychev Laboratory of St. Petersburg State University was successful. Unfortunately I cannot access this funding anymore, as I am leaving Berlin for Pisa....