The numerical minimization of the functional $\mathcal{F}\left(u\right)={\int }_{\mathrm{\Omega }}\varphi \left(x,{\nu }_u\right)\left|Du\right|+{\int }_{\partial \mathrm{\Omega }}\mu ud{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}\kappa udx$, $u\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$ is addressed. The function $\varphi$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $\mathcal{F}$ can be equivalently minimized on the convex set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and then regularized with a sequence ${\left\{{\mathcal{F}}_{ϵ}\left(u\right)\right\}}_{ϵ}$, of stricdy convex functionals defined on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$, can be discretized by continuous linear finite elements. The convexity property of the functionals on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful in the numerical...

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results...

This Note is the Part II of a previous Note with the same title. One refers to holonomic systems $\mathrm{\Sigma }=\mathcal{A}\bigcup \mathcal{U}$ with two degrees of freedom, where the part $\mathcal{A}$ can schemetize a swing or a pair of skis and $\mathcal{U}$ schemetizes whom uses $\mathcal{A}$. The behaviour of $\mathcal{U}$ is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on $\mathrm{\Sigma }$ minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition $\mathrm{\Gamma }$ which...

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.

In some preceding works we consider a class $\mathcal{O}\mathcal{P}$ of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line $l=l_{\gamma (\cdot )}$, regarded as determined by its (variable) curvature function $\gamma (\cdot )$ of domain $\left[s_0,s_1\right]$. Assume that the problem $\tilde{\mathcal{P}}\in \mathcal{O}\mathcal{P}$ is regular but has an impulsive monotone character in the sense that near each of some points ${\delta }_1$ to $\delta {}_{\nu }\gamma (\cdot )$ is monotone and $\left|\gamma {}^{\prime }\right(\cdot \left)\right|$ is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify $\tilde{\mathcal{P}}$ into a structurally...