Gibson 1st Floor SR

Defect measures have successfully been used, in a variety of

contexts, as a tool to quantify the lack of compactness of bounded

sequences of square-integrable functions due to concentration and

oscillation effects. In this talk we shall present some results on the

structure of the set of possible defect measures arising from sequences

of solutions to the linear Schrödinger equation on a compact manifold.

This is motivated by questions related to understanding the effect of

geometry on dynamical aspects of the Schrödinger flow, such as

dispersive effects and unique continuation.

It turns out that the answer to these questions depends strongly on

global properties of the geodesic flow on the manifold under

consideration: this will be illustrated by discussing with a certain

detail the examples of the the sphere and the (flat) torus.

Self-gravitating elastic bodies provide models for extended

objects in general relativity. I will discuss constructions of static

and rotating self-gravitating bodies, as well as recent results on the

initial value problem for self-gravitating elastic bodies.

I will report my work on rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, in which we obtain the local well-posedness result in $H^s$, $s$>$2$. The novelty of this approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\bf{g}$, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $\Box_{\bf{g} } \phi=0$ directly. If time allows, I will talk about my work in progress on the sharp results for the more general quasilinear wave equations by vector fields approach.

The talk will discuss generalizations of the classical Hadamard jump  condition to general locally Lipschitz maps, and applications to
polycrystals. This is joint work with Carsten Carstensen.

Let ${T_1,...,T_l}$ be a collection of differential operators

with constant coefficients on the torus $\mathbb{T}^n$. Consider the

Banach space $X$ of functions $f$ on the torus for which all functions

$T_j f$, $j=1,...,l$, are continuous. The embeddability of $X$ into some

space $C(K)$ as a complemented subspace will be discussed. The main result

is as follows. Fix some pattern of mixed homogeneity and extract the

senior homogeneous parts (relative to the pattern chosen)

${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there

are two nonproportional operators among the $\tau_j$ (for at least one

homogeneity pattern), then $X$ is not isomorphic to a complemented

subspace of $C(K)$ for any compact space $K$.

The main ingredient of the proof is a new Sobolev-type embedding

theorem. It generalises the classical embedding of

${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that

now the integrability condition is imposed on certain linear combinations

of derivatives of different order of several functions rather than on the

first order derivatives of one function.

This is a joint work with D. Maksimov and D. Stolyarov.

In this talk we show how degree N maps of the form $u_{N}(z) = \frac{z^{N}}{|z|^{N-1}}$ arise naturally as stationary points of functionals like the Dirichlet energy. We go on to show that the $u_{N}$ are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N. Meyers in the 60s, and another where the constraint $\det \nabla u = 1$ a.e. plays a prominent role.

The proof of several properties of solutions of hyperbolic systems of conservation laws in one space dimension (existence, stability, regularity) depends on the existence of a decreasing functional, controlling the nonlinear interactions of waves. In a special case (genuinely nonlinear systems) the interaction functional is quadratic, while in the general case it is cubic. Several attempts to prove the existence of a a quadratic functional also in the most general case have been done. I will present the approach we follow in order to prove this result, an some of its implication we hope to exploit.

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Work in collaboration with Stefano Modena.

The new schedule will follow shortly

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all

models of turbulence. In particular, K41 and corrections to it provide

estimates of small-scale quantities such as increments and energy spectrum

for a 3D turbulent flow. However, because of the well-known difficulties

involved in studying 3D turbulent flow, there are no rigorous results

confirming or infirming those predictions. Here, we consider a well-known

simplified model for 3D turbulence: Burgulence, or turbulence for the 1D

Burgers equation. In the space-periodic case with a stochastic white in

time and smooth in space forcing term, we give sharp estimates for

small-scale quantities such as increments and energy spectrum.

We first provide a brief overview of some of the key properties of the space $\textrm{BV}(\Omega;\mathbb{R}^{N})$ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral

\[

F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,}

\]

where $\Omega\subset\mathbb{R}^{n}$ is open and bounded, and $f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is a continuous function satisfying the growth condition $0\leq f(\xi)\leq L(1+|\xi|^{r})$ for some exponent $r$. When $u\in\textrm{BV}(\Omega;\mathbb{R}^{N})$, we extend the definition of $F(u;\Omega)$ by introducing the functional

\[

\mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left|

\!\!\begin{array}{r}

(u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\

u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N})

\end{array} \right. \bigg\} \,\textrm{.}

\]

\noindent For $r\in [1,\frac{n}{n-1})$, we prove that $\mathscr{F}$ satisfies the lower bound

\[

\mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,}

\]

provided $f$ is quasiconvex, and the recession function $f_{\infty}$ ($:= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $r=1$; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.