I focus on the analysis of nonlinear PDEs and of variational problems that arise in models from physics. My objectives are to understand well-posedness of these models as well qualitative properties of solutions. I have been working on models from Fluid Dynamics, Ferromagnetism, Elasticity and Diblock-Copolymers. 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 line
- Ferromagnetic samples
- Elastic materials
- Other variational models
Fluid evolution in the presence of a moving contact line
Consider 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 no-slip 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 well-posedness 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 thin-film equations. Together with L. Giacomelli and F. Otto, I have established a well-posedness theory for certain thin-film equations.
- L. Giacomelli, H.K., F. Otto, Smooth solutions for a thin-film equation around the stationary solution,
J. Differential Equations, 245 no 6, pp 1454-1506, 2008. article preprint abstract
- H.K., L. Giacomelli, A free boundary problem of fourth order: Classical solutions in weighted Hölder spaces
Comm. Part. Diff. Eq. 35 no 11, pp 2059-2091, 2010. article preprint abstract
- H.K., Well-posedness for the Navier slip thin-film equation in the case of partial wetting,
Comm. Pur. Appl. Math. 64, pp 1263-1296, 2011. article preprint abstract
- H.K.,Well-posedness for a class of thin-film equations with general mobility in the regime of partial wetting,
submitted. preprint abstract
Together with Nader Masmoudi, I have also analyzed the rigorous lubrication approximation: We start with a 2-d fluid model where the fluid evolution is governed by Darcy’s Law. In the limit of thin films, we show that solutions converge to solutions of a certain thin-film equation
- H.K., N. Masmoudi, Well-posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge,
Comm. Math. Phys., accepted 2012. preprint abstract
- H.K., N. Masmoudi, Darcy flow on a plate with prescribed contact angle: Well-posedness and lubrication approximation,
preprint. preprint abstract
The 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, non-convex and non-local. 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.
- A. DeSimone, H.K., F. Otto, 2-d stability of the Néel wall,
Calc. Var. Partial Differential Equations 27 no 2, pp 233-253, 2006. article preprint abstract
- R. Ignat, H.K., Vortex energy and 360°-Néel walls in thin-film micromagnetics,
Comm. Pur. Appl. Math 63 no 12, pp 1677-1724, 2010. article preprint abstract
- H.K., C. Muratov, Domain structure of bulk ferromagnetic crystals in applied fields near saturation,
J. Nonlin. Sc., accepted in 2011. article preprint abstract
Shape 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 non-convex. These two characteristics lead to a rich pattern formation.
- H.K., R.V. Kohn, Minimal energy for elastic inclusions,
Proc. Royal Soc. Lond. A 467 no 2127, pp 695-717, 2011. article preprint abstract
- H.K., R.V. Kohn, F. Otto, Nucleation barriers for the cubic-to-tetragonal phase transformation,
Comm. Pur. Math. Appl. , accepted in 2011. article preprint abstract
Other variational models
The 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 long-range force. We have first focused on the 2-d 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 non-existence of minimizers (regularity has been established before). We also show that the disc is the exact minimizer for small masses. Furthermore, for certain long-range 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.
- H.K., C. Muratov, On an isoperimetric problem with a competing non-local term. I. The planar case,
Comm. Pur. Appl. Math., accepted 2012 . article preprint abstract
In a new work, we have extended some of the 2-d results to the case of higher space dimensions (smaller than 8). In the case, when the model is characterized by a certain long-range 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.