We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_0\in X$, where $X\in \{M_{2,q}^s\left(ℝ\right),H^{\sigma }\left(𝕋\right),H^{s_1}\left(ℝ\right)+H^{s_2}\left(𝕋\right)\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s\left(ℝ\right)\hookrightarrow L^3\left(ℝ\right)$, or if $\sigma \ge \frac{1}{6}$, or if $s_1\ge \frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...

The study of $J$-holomorphic maps leads to the consideration of the inequations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, and $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of $u$ vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Hölder class $\frac{1}{2}$, any $J$-holomorphic curve that is constant on a non-empty...

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere ${𝕊}^n$ of ℝn+1. In this paper, we mainly consider the uniqueness question and give first results.