We prove the existence and uniqueness of global solutions to the mixed problem for the Carrier equation $$ u_{tt} - M(\int_{\Omega} u^{2}\, dx ) \Delta u + g(u_{t}) = f, $$ where $g^{\prime}(s) \geq 0, \, 0 < m_{0} \leq M(\lambda)$ and no "smallness" conditions are imposed on the initial data. Moreover, the algebraic and exponential decays of the energy are proved.

Results of the type of Filippov's Theorem are proved for nonconvex differential inclusions whose state variable is constrained in the closure of an open subset of $\mathbb{R}^n$. An application is provided to dynamic programming for optimal control problems with state constraints and a control set depending on the time and the state.

In this paper we continue the analysis, started in [6],[7], of the regularity of solutions of free discontinuity problems. We choose as a model problem the minimization of the Mumford-Shah functional. Assuming that in some region the optimal discontinuity set $\Gamma$ is the graph of a $C^{1,\rho}$ function, we look for conditions ensuring the higher regularity of $\Gamma$. Our results are optimal in the two dimensional case. As an application, we prove that in the case of the Mumford-Shah functional and in similar problems the Lavrentiev phenomenon does not occur.

In this paper it is proven the existence of light--like geodesics joining an event $p$ and a time--like vertical curve $\gamma$ of a Lorentzian manifold $\mathcal{M}$ endowed with a Universal Time Function $T$, under a certain compactness condition. Moreover, it is developed a Ljusternik--Schnirelman theory for light rays, using which it is shown that, if the topology of $\mathcal{M}$ satisfies a non--triviality condition, then there are multiple light rays joining $p$ with $\gamma$. The results are obtained under intrinsic assumptions on the manifold $\mathcal{M}$, that do not involve the coefficients of the Lorentzian metric

If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq 3$ or if $f$ is given by $e^u$ when $N=3$, we prove the existence of a positive weak solution of $ \Delta u + \lambda f(u) =0 $ which is defined in the unit ball of ${\Bbb R}^N$, has $0$ boundary data and has a nonremovable prescribed singularity at some point $x_0$ close to the origin.

We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to $\varepsilon$ we prove that an immediate exchange of stabilities implies a change of the asymptotic behavior from $0(\varepsilon)$ to $0(\sqrt{\varepsilon})$ near the point of exchange of stabilities.

The main purpose of this paper is to show that all nonnegative solutions to $ \Delta u = 0$ (or $ \Delta u = u^p$) in the n-dimensional upper half space $H = \{ (x', t)| x'\in \Bbb{R}^{n-1}, t>0 \}$ with boundary condition $ {\partial u}/{\partial t}= u^q $ on $\partial H$ must be linear functions of $t$ (or $u\equiv 0$) when $n \ge 2$ and $q>1$ (or $n\ge 2$ and $p, q>1$).