Gibson 1st Floor SR

Several dissipative scalar conservation laws share the properties of

$L1$-contraction and maximum principle. Stability issues are naturally

posed in terms of the $L1$-distance. It turns out that constants and

travelling waves are asymptotically stable under zero-mass initial

disturbances. For this to happen, we do not need any assumption

(smallness of the TW, regularity/smallness of the disturbance, tail

asymptotics, non characteristicity, ...) The counterpart is the lack of

a decay rate.

We consider monochromatic planar electro-magnetic waves propagating through a nonlinear dielectric medium in the optical regime.

Travelling waves are particularly simple solutions of this kind. Results on the existence of guided travelling waves will be reviewed. In the case of TE-modes, their stability will be discussed within the context of the paraxial approximation.

The talk starts with the observation that many well-known systems of diffusive type

can be written as Wasserstein gradient flows. The aim of the talk is

to understand _why_ this is the case. We give an answer that uses a

connection between diffusive PDE systems and systems of Brownian

particles, and we show how the Wasserstein metric arises in this

context. This is joint work with Johannes Zimmer, Nicolas Dirr, and Stefan Adams.

Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria.

We further discuss the scaling laws and $\Gamma$-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric on a bounded domain into $\mathbb R3$.

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).

Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces

for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients.

The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator

and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO)

which guarantee that the solution operator associated with this problem is an isomorphism.

We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist.