| Frank Duzaar: $p$-Harmonic Approximation |
| Abstract: The main part of the talk will be concerned with the
so called $p$-harmonic approximation lemma, $p>1$. The lemma states
that a given map $u\in W^{1,p}(B,\mathbb{R}^N)$ ($B$ the unit ball
in $\mathbb{R}^n$, $N\ge1$) with bounded $p$-energy $\int_B |Du|^p\,
dx\le 1$ can be approximated in the $L^p$-norm by some $p$-harmonic
function, i.e.\ an exact solution $h\colon B\to\mathbb{R}^N$ of the
$p$-Laplacian system
$$
\sum_{\alpha=1}^n D_\alpha\left( |Du|^{p-2}D_\alpha u\right)
=0 \quad\mbox{on $B$,}
$$
provided $u$ is approximatively $p$-harmonic in a certain sense.
For $p=2$ the lemma reduces to the well-known blow-up lemma.
At the end of the talk we will discuss applications of the $p$-harmonic
approximation lemma to the regularity problem for $p$-harmonic maps and
for minimizers of degenerate quasi-convex variational integrals with
$p$-growth. All presented results are joint work with G. R. Mingione
from Parma. |
| Mariano Giaquinta: About limits of sequences of maps into
Riemannian manifolds with equibounded energies.
|
| Abstract: The aim of the talk is to report on recent and less recent results
concerning the characterization of limit points of smooth maps with
values into a manifolds in the contest of geometric measure theory and,
in particular, in the contest of Cartesian currents. Specially, we shall
discuss the case of Dirichlet energy. |
| Qing Han: Harmonic functions and several complex variables |
| Abstract: Traditionally, an important tool to study harmonic functions in $R^2$
is to identify $R^2$ as $C$ and identify harmonic functions as real parts of
analyic functions. In this talk, I will present another method. I will consider
$R^2$ as a part of $C^2$ and harmonic functions as the restriction on $R^2$ of
holomorphic functions in $C^2$. By this method, we study singular sets of harmonic
functions. Here singular sets are the critical zero sets. They are the singular
part of zero sets of harmonic functions. For harmonic functions in $R^2$, the
singular set is isolated. In fact, the complex singular set is also isolated
for the holomorphic functions in $C^2$ extended from the harmonic functions.
We are interested in the number of singular points both in $R^2$ and $C^2$.
We shall show that both quantities can be estimated in terms of frequency of
harmonic functions.
|
| Vagn Lundsgaard Hansen: The component problem in mapping spaces |
| Absract: The components in a space of continuous maps in general display
a surprisingly rich variety of different topological types, reflecting the
existence of an interesting basic Morse theory in such infinite dimensional
manifolds. The division into homotopy types of the components in a fixed
mapping space was formally introduced as {\em the component problem}
in mapping spaces in the beginning of the 1970s. In this lecture we shall
discuss the component problem and some of the results obtained. In rare
cases, the full homotopy type of a component in a mapping space can be
determined explicitely in terms of well understood spaces. As an example,
the homotopy type of the identity component in the space of maps on the
2-sphere has been fully determined, whereas the case of the homotopy type
of the component of constant maps is still an open problem. |
| Tadeusz Iwaniec: Weakly differentiable mappings between manifolds |
| Abstract: We study Sobolev classes of weakly differentiable mappings
between compact Riemannian manifolds without boundary. The novelty of our
results is that we do not impose any topological conditions on the
manifolds. The main topics to be discussed are: a) Approximation by smooth
mappings, b) Integrability of the Jacobian determinant. We characterize
(essentially all) Sobolev type classes in which smooth mappings are dense.
These mappings turn out to be continuous when restricted to certain tiny sets.
These sets give rise to the so-called web like structure associated with the given
mapping. The integrability properties of the Jacobian determinants in a manifold
setting are really different than one might a priori expect based on the well
known results in the Euclidean space. To our surprise, the case when the target
manifold admits only trivial l-cohomology groups, with 0 < l < n (like n-sphere),
is more difficult than the nontrivial case in which the target manifold has at
least one non-zero l-cohomology, 0 < l < n. The necessity of the nontrivial
cohomology in the target manifold is a new phenomenon in the regularity theory
of the Jacobians. |
| Bernd Kirchheim: Multi-well problems and shape memory materials |
| Abstract: We discuss the differential inclusions describing the zero energy
state for certain shape memory alloys and discuss attempts and ideas to construct
solutions for this inclusion. An aim is to present recent progress in the question
which global behaviour in 3 dimensions allows almost energy free microstructures and
the construction of particular simple solutions for a 2 dimensional model problem. |
| Jan Kristensen: Uniform and mean oscillation estimates for minimizers
of multi-well energies
|
| Abstract: For compact sets $K \subset {\bf R}^{2 \times 2}$ of two by two
matrices we consider the corresponding multi-well energies
$$
I[u] = \int_{\Omega} \! \mbox{dist}^{2}^ (\nabla u(x),K) \, dx,
$$
defined for Sobolev maps $u \colon \Omega \subset {\bf R}^{2} \to
{\bf R}^{2}$. The topic of this talk is local $L^{\infty}$ and
$BMO$ estimates for the derivatives $\nabla \bar{u}$ of minimizers
$\bar{u}$ of $I[u]$ and its relaxation $\bar{I}[u]$. The local $BMO$
estimate holds in general and is a consequence of standard estimates
for the Laplacian. The question of whether or not the local $L^{\infty}$
estimate holds is more subtle. In general it does not, and this is
connected to the well-known phenomenon that certain singular integrals
do not map $L^{\infty}$ into itself.
Parts of the talk are based on joint work with Georg Dolzmann
(Maryland, USA) and Kewei Zhang (Sussex, UK). |
| Sergei Kuksin: Perturbed harmonic map equations and elliptic PDEs on
manifolds |
| Abstract: I shall discuss elements of a theory for elliptic equations for
maps between manifolds. The theory applies to the harmonic maps
equations for maps to negatively-curved manifolds, and allows to count
algebraical number of solutions for these and similar equations. |
| Ernst Kuwert: Removability of point singularities of Willmore surfaces |
| Abstract: This is a report on joint work with Reiner Schatzle (Bonn).
Willmore surfaces with isolated singularities and square integrable
curvature come up in blowup constructions for sequences of Willmore
surfaces and for the corresponding gradient flow, the Willmore flow.
We show that unit density, isolated singularities in codimension one
are removable under some extra condition. As an application we obtain
the following optimal result: the Willmore flow of any sphere with
energy not exceeding $8\pi$ converges smoothly to a round sphere.
We also deduce a compactness result for Willmore tori with energy
below $8 \pi$. |
| Yanyan Li: On some conformally invariant fully nonlinear equations: Liouville,
Yamabe and Harnack |
| Abstract: I present some recent work on some conformally invariant fully nonlinear
equations. This includes works on Liouville type theorems, a fully nonlinear version of
the Yamabe problem on locally conformally flat manifolds and some Harnack type inequalities. |
| Jan Maly: Fine properties of Sobolev functions related to the co-area formula |
| Abstract: A co-area formula for Sobolev transformation of variables has been recently
obtained (a joint work with D. Swanson and W. P. Ziemer).This, and related Eilenberg-type
inequality, rely on "capacitary"estimates of $m$-codimensional Hausdorff content of level
sets. The sharp assumption is that the gradient of the transformation belongs to the Lorentz
space $L_{m,1}$. Then we also obtain that the set of non-Lebesgue points has $m$-codimensional
Hausdorff measure zero. The key estimates can be performed in the framework of metric measure
spaces.
|
| Paul Rabinowitz: On some results of Moser and of Bangert |
| Abstract: Moser and then Bangert took some steps towards PDE analogues
of the work of Aubry and Mather on monotone twist maps. We
will describe recent results with Stredulinsky in the same
direction that were motivated by phase transition problems. |
| Tristan Riviere: The singular set of almost complex cycles |
| Jeyabal Sivaloganathan: Singular weak solutions and modelling fracture
in nonlinear elasticity |
| Abstract: In this talk we study existence and properties of singular weak solutions to
the equations of nonlinear elasticity. Using these, we propose a variational model for the
initiation of a fracture/crack in an elastic material. This is joint work with S.J. Spector
(S. Illinois,USA). |
| Michael Struwe: Convergence of the Yamabe flow for `large' energies |
| Charles Stuart: Global bifurcation using the degree for Fredholm maps |
| Abstract: The topological degree for proper C1- Fredholm maps of index zero provides a
good tool for establishing global bifurcation of solutions decaying to zero
at infinity of quasilinear elliptic equations on RN. I shall summarise my
recent work with Patrick Rabier in this direction. |
| Hans Triebel: The fractal Laplacian; multifractal quantities |
| Abstract: Let $\mu$ be a finite Radon measure in the plane $\R^2$ such that its support $\Gamma$
is compact and has Lebesgue measure zero. Let $\mu_j = \sup_{m \in \Z^2} \mu (Q_{jm}), j \in
\N_0$ where $Q_{jm}$ is the canonical tiling of $\R^2$by squares with the side length
$2^{-j}$. Let $\Omega$ be a bounded $C^{\infty}$ domain in $\R^2$ with $\Gamma \subset \Omega$,
and let $-\Delta$ be the Dirichlet Laplacian with respect to $\Omega$. If $\{ \mu_j \} \in
l_{\frac{1}{2}}$, then the operator $B=(-\Delta)^{-1} \circ \mu$ is compact, self-adjoint,
non-negative in $H^1_0 (\Omega)$. The well-known classical Courant property (Courant 1924)
has the following fractal counterpart: The largest eigenvalue is simple, the related eigenfunction
$u$ (Courant function) is continuous in $\bar{\Omega}$, positive in $\Omega$ (up to a constant),
harmonic in $\Omega \backslash \Gamma$, and reflects faithfully global and local multifractal
quantities of $\mu$ in terms of the Besov spaces $B^{\sigma}_p (\Omega)=B^{\sigma}_{pp} (\Omega)$.
For example, the Courant characteristic of $\mu$, $\omega_{\mu}(t) = \sup \{ \omega : u \in
B^{\omega}_p (\Omega) \}$ are related to multifractal quantities by $\omega_{\mu}(t) = 2 t -
t \limsup_{j \to \infty} \frac{1}{j} \log \left( \Sigma_{m \in \Z^2} \mu(Q_{jm})^{\frac{1}{t}}
\right)$, if $0 < t < 1$ (log taken to base $2$). But these distinguished eigenfunctions $u$ have
a much deeper sophisticated knowledge about $\mu$ (waiting to be asked properly: a fractal
oracle). |