



Research InterestsI focus on the analysis of nonlinear PDEs and of variational problems that arise in models from physics. My objectives are to understand wellposedness of these models as well qualitative properties of solutions. I have been working on models from Fluid Dynamics, Ferromagnetism, Elasticity and DiblockCopolymers. In the following, I give an overview of my main goals in the specific areas I have worked on:
Fluid evolution in the presence of a moving contact lineConsider a spreading droplet on a solid substrate. The contact line denotes the place where the three phases (air, liquid, solid) meet. Surprisingly, the classical assumptions of fluid dynamics do not allow for the propagation of the contact line. This observation has been first announced by Huh and Scriven in 1971. Different (still highly discussed) models have been introduced in the physical community to avoid the noslip paradox. Since the singularity set of the contact line is experimentally as well numerically hard to investigate, it is still not clear which of these models describes the physics best. This leads to many interesting mathematical questions: One question is to ask about wellposedness of fluid models in the presence of a contact line. Another question I'm interested in is the qualitative behaviour of solutions near the contact lines. A convenient to analyze moving contact is given by the family of thinfilm equations. Together with L. Giacomelli and F. Otto, I have established a wellposedness theory for certain thinfilm equations.
Ferromagnetic samplesThe stable states of the magnetization of a ferromagnetic body can be characterized as the local minima of the micromagnetic energy functional, introduced in 1935 by Landau and Lifschitz. Ferromagnetic materials are of huge interest in many applications such as e.g. the storage of bonary information in hard disc or in MRAM devices. The micromagnetic functional is in particular challenging since it is vectorial, nonconvex and nonlocal. Together with coworkers, I have analyzed the ground state energy and low energy patterns as well for bulk ferromagnetic samples as well as for thin ferromagnetic films.
Elastic MaterialsShape memory alloys are characterized by a preferred crystal lattice structure for high temperatures (austenite) and a preferred crystal lattice structure for low temperatures. Generally, the austenite lattice has a higher symmetry than the martensite lattice. Therefore, there exist several symmetry related Martensite lattice structures. The corresponding variational model is tensorial and nonconvex. These two characteristics lead to a rich pattern formation.
Other variational modelsThe isoperimetric problem is a classical problem in the calculus of variations, one formulation of which seeks to find a set of the smallest perimeter enclosing a prescribed volume. By the famous result of De Giorgi, in the Euclidean space the solution of this problem is well known to be a ball. In nature, often also nonlocal interactions (such as Coulomb interactions) play an important role in pattern formation. Together with Cyrill Muratov, I'm working on the question how the solution of the isoperimetric problem is affected by an addition of a repulsive longrange force. We have first focused on the 2d dimensional case in order to avoid some technical difficulties associated to higher dimensions. We have a quite complete picture of the variational problem, including existence and nonexistence of minimizers (regularity has been established before). We also show that the disc is the exact minimizer for small masses. Furthermore, for certain longrange type interactions we are able to completely solve the varitional problem: The minimizer is a ball up to a certain critical mass and it does not exist for higher masses.
In a new work, we have extended some of the 2d results to the case of higher space dimensions (smaller than 8). In the case, when the model is characterized by a certain longrange interactions, we show that existence and nonexistence of minimizers (depending on the mass). Furthermore, we show that the exact minimizer is given by the ball.
