Stationäre mikropolare elektrorheologische Fluide - Existenz schwacher Lösungen bei Spannungstensoren mit voneinander verschiedenen Wachstumseigenschaften
Dienstag, 27.4.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
Es wird die Existenz schwacher Lösungen des stationären Systems, das die Bewegung mikropolarer elektrorheologischer Fluide beschreibt, auf dreidimensionalen Gebieten untersucht. Dabei wird auf die Theorie pseudomonotoner Operatoren zurückgegriffen. Darüber hinaus wird der Ansatz der Lipschitz-Truncation benötigt, um die Existenz für kleine Scherungsexponenten (> 6/5) zu zeigen. Besonders ist hier, dass Spannungstensor und Momentenspannungstensor mit voneinander verschiedenen Wachstumseigenschaften versehen sind, wodurch sich einige Zusatzvorausssetzungen ergeben.
Variational models for line-defects in materials.
Dienstag, 4.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
The purpose of the seminar is to describe a fully 3d model for dislocations derived by the asymptotic analysis of geometrically nonlinear elastic energy with quadratic growth. Precisely we obtain, through a Γ-convergence result, that the energy stored by a distribution of dislocations in a crystal is the contribution of a volume term representing the elastic energy and a line tension term representing the plastic energy.
On the minimization of various energies of Riesz-type.
Dienstag, 11.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
\n\n\nThe celebrated liquid drop model by Gamow, which dates back to the 1930's, has attracted since then a lot of attention among physicists and mathematicians. In particular, there has been a deep increase in the mathematical interest about this problem and several generalisations in the last decade, with many proven results but also fundamental questions still open. We will give a general overview on these problems, concluding with some very recent results, obtained in some collaborations with Carazzato, Fusco, Novaga.\n\n
Taylor Scaling for curvature driven interfaces in random media
Dienstag, 18.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
We present a model for curvature driven interface propagation through a homogeneous medium with random obstacles. The energy is fully nonlinear and the dissipation is mixed, capturing both viscous dissipation as well as dry friction. If the interface passes over an obstacle it incurs additional dry friction. This model is relevant for the study of dislocations. Under an applied force, we investigate the pinning (i.e., a solution becomes stuck) and depinning behavior of the interface. We show that the model obeys Taylor Scaling, i.e., the critical pinning force scales like the square root of the concentration of the obstacles. Joint work with Patrick Dondl (Freiburg) and Michael Ortiz (Pasadena).
Homogenization of second order level set PDE in periodic media
Dienstag, 1.6.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
I will discuss the homogenization of a class of interface motions "in nondivergence form" with periodically varying coefficients. Compared to uniformly elliptic second order PDE, the difficulty here is the equation only induces averaging in d - 1 dimensions. This leads to strong anisotropic effects: in particular, typically the homogenized coefficients are discontinuous. I will describe the behavior of the homogenized coefficients at discontinuities and also explain how to prove a comparison principle for the effective motion.
Homogenization in a class of non-periodically perforated domains
Dienstag, 15.6.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
We consider the deterministic homogenization of the Poisson problem and the Stokes system in a class of non-periodically perforated domains. The size of the perforations is comparable to the distance between two neighbouring holes. The boundary conditions for both problems are of homogeneous Dirichlet type along the holes and the macroscopic boundary. The homogenization of these PDEs when the holes are periodically distributed in space is well-known. We aim at extending these results to local perturbations of the periodic case, that is when the geometry is not periodic but tends to be periodic far from the origin. This setting takes into account local defects that could appear in a pure periodic microstructure. In this talk, we first introduce the conditions imposed on the non-periodic porous medium. We then construct classical objects of the homogenization such as correctors and we obtain convergence rates of the solution to its two scale expansion for both Poisson problem and Stokes system. We finally comment on the optimality of these convergence rates.\n
Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions
Dienstag, 13.7.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
\n\nWe show uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard’s algorithm. We show how equations of this type naturally appear in models of mathematical finance.
Variable exponent Bochner-Lebesgue spaces with symmetric gradient structure
Dienstag, 20.7.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
We introduce function spaces for the treatment of non-linear parabolic equations with variable log–Hölder continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analog of Korn’s inequality for these functions spaces is not available, the construction of an appropriate smoothing method proves itself to be difficult. Using a pointwise Poincaré inequality near the boundary of a bounded Lipschitz domain\ninvolving only the symmetric gradient, we construct a smoothing operator with convenient properties. In particular, this smoothing operator leads to several density results, and therefore to a generalized formula of\nintegration by parts with respect to time. Using this formula and the theory of maximal monotone operators, we prove an abstract existence result."