Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle,$$i{\partial }_{t}u={\Pi \left(\right|u|}^{2}u),$$where $\Pi$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...

We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the...

Si prova l'esistenza di un'unica soluzione debole che dipende con continuità dai dati al contorno per il problema lineare di Molodenskii in approssimazione quasi sferica, nel caso che la superficie al contorno soddisfi una condizione di cono. Si segue un approccio costruttivo diretto, che generalizza una procedura precedentemente elaborata per il problema semplice di Molodenskii. Inoltre si prova che la soluzione ha derivate prime a quadrato integrabile al contorno, il che è essenziale per le applicazioni...

Si studiano le condizioni per 1’esistenza, l’unicità e la stabilità della soluzione debole del problema lineare di Molodenskii in approssimazione quasi-sferica, generalizzando una tecnica perturbativa usata in precedenza per la soluzione di tipo classico. La procedura seguita richiede delle condizioni di maggior regolarità per il contorno, di quelle usate nell’analisi del problema «semplice». Il risultato ottenuto è l'esistenza e unicità di una soluzione con derivate seconde a quadrato integrabile,...

We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta u+u+{\left(\right|x|}^{-4}\ast \left|u\right|²)u=0$ in spatial dimension d ≥ 5. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the momentum vector...

A proof is given of the following theorem: infinitely differentiable solenoidal vector - functions are dense in the space of functions, which are solenoidal in the distribution sense only. The theorem is utilized in proving the convergence of a dual finite element procedure for Dirichlet, Neumann and a mixed boundary value problem of a second order elliptic equation.

We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on $H^m\left({ℝ}^{\nu }\right)$. The support of the density coincides with the spectrum of the operator in $L²\left({ℝ}^{\nu }\right)$.

Let $u(x,y)$ be a harmonic function in the half-plane ${\mathbf{R}}_{+}^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r),-\infty \>r\>\infty$, that are analogs of the family of local times associated to the process $u(x_t,y_t)$ where $(x_t,y_t)$ is Brownian motion in ${\mathbf{R}}_{+}^{n+1}$. We show that $D\left(u\right)={sup}_{r}D(u;r)$ is bounded in $L^p$ if and only if $u(x,y)$ belongs to $H^p$, an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.