### 5-Shaped Bifurcation Curves of Nonlinear Elliptic Boundary Value Problems.

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In questo lavoro si considera un’equazione alle derivate parziali del primo ordine con una condizione sulla frontiera di tipo integrale. Si studia resistenza, l'unicità e il comportamento asintotico delle soluzioni.

The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution is defined via a variational inequality, following Temam?s technique for the evolution problem...

In this addendum we address some unintentional omission in the description of the swimming model in our recent paper (Khapalov, 2013).

We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in ${\mathbb{R}}^{d}$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in ${\mathbb{R}}^{3}$. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma}_{T}$, $\mathbb{P}\left({\Sigma}_{T}\right)>0$ such that any data ${\phi}^{\omega}\left(x\right)\in {H}^{\gamma}\left({\mathbb{T}}^{3}\right),\gamma <1,\omega \in {\Sigma}_{T}$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-{e}^{it\Delta}{\phi}^{\omega}\in C([0,T];{H}^{s}\left({\mathbb{T}}^{3}\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space ${H}^{1}\left({\mathbb{T}}^{3}\right)$, that is in the supercritical scaling regime.