Suppose $u$ satisfies the equation $\Delta u + F(x,u)$ on $\Omega$. When the domain is deformed through a one-parameter family $\{\Omega_t\}$, the Cauchy data of $u$ on $\partial\Omega_t$ will satisfy a first-order evolution equation. This equation is ill-posed, in the sense that it does not admit solutions forwards or backward in time for generic initial data. However, if the domain is deformed smoothly to a point, this equation admits an exponential dichotomy, so there exist two distinguished subspaces of boundary data (at each time $t$) for which solutions exists forward and backward in time, respectively. When the PDE is selfadjoint, the evolution equation will be Hamiltonian, so the unstable subspace is Lagrangian and hence has a well defined Maslov index. These constructions generalize previous work in spatial dynamics, which considered elliptic equations on cylindrical domains.

This is joint with with Margaret Beck, Christopher Jones, Yuri Latushkin and Alim Sukhtayev.

\begin{equation} - {\rm div} (a(x) D u) + M \, \displaystyle\frac{|D u|^2}{u^{\theta}}= f(x) \,\,\,\, \mbox{in $\Omega$} \tag{1} \end{equation} where $\Omega $ is an open bounded subset of $\mathbb{R}^N$ $(N \geq 3)$ with smooth boundary, $a(x)$ is a $L^{\infty}$--matrix satisfying the standard ellipticity condition, $\theta \in ]0,1[$, $M$ is a positive constant and $f$ is sufficiently regular i.e. it belongs to a suitable Morrey space.

Namely, we assume that the right-hand side $f$ belong to the Morrey space $L^{m, \lambda}(\Omega)$ with \[ 1 \leq m \leq \displaystyle\frac{2 N}{2N-\theta(N -2)} \,\,\, \mbox{ and} \,\,\, 0 < \lambda < N-2 \] so that our right-hand side doesn't belong to the natural dual space or it is "nearly" a measure ($m=1$).

We will be concerned with the regularity of the gradient of a solution in Morrey spaces in correspondence with the Morrey properties of the right--hand side of the equation (1).

We will start to show the result due to Escobar and independently to Beckner: there exists a positive constant $K_n$, depending only on $n$, such that for any function $f\in W^{1,2}(\mathbb{R}^{n}_{+})$, it holds $$(\int_{\mathbb{R}^{n-1}} |f(0,x)|^{\frac{2(n-1)}{n-2}} dx)^{\frac{n-1}{n-2}}\leq\ K_n \int_{\mathbb{R}^{n}_{+}} |\nabla f(t,x)|^2 dxdt,$$ where $(t,x)\in \mathbb{R}_{+} \times \mathbb{R}^{n-1}=\mathbb{R}^{n}_{+}$.

Afterwards we will consider the case of fractional Sobolev inequalities, presenting the fractional counterpart of the previous statement and some possible generalizations. (This is a work in progress with Jie Xiao)

Joint work with K. Kellay.

Xiaomin Tang

Huzhou University

In this talk, we first establish a new type of the classical Schwarz lemma at the boundary for holomorphic self-mappings of the unit ball in $\mathbb{C}^n$, and then give the boundary version of the rigidity theorem. This is a joint work with Taishun Liu and Wenjun Zhang.

My presentation is about the general dual extension of the classical Minkowski problem---the general dual Orlicz-Minkowski problem. That is, {\em for which nonzero finite Borel measures $\mu$ on $S^{n-1}$ and continuous functions $G$ and $\psi$ do there exist a constant $\tau\in \mathbb{R}$ and a convex body $K$ such that $\mu=\tau\,\widetilde{C}_{G, \psi}(K,\cdot)$?} Here $\widetilde{C}_{G, \psi}(K,\cdot)$ is the finite signed Borel measure. In particular, a solution to this problem will be presented. This talk is based on a joint work with Richard Gardner, Daniel Hug, Wolfgang Weil and Deping Ye.