In this paper we consider two-dimensional quasilinear equations of the form $\text{div}\left(a\left(\left|\nabla u\right|\right)\nabla u\right)+f\left(u\right)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u$(notice that $\text{arg}\left(\nabla u\right)$ is a well-defined real function since $\left|\nabla u\right|>0$ on ${\mathbb{R}}^2$) we prove that $u$ is one-dimensional, i.e., $u=u\left(\nu \cdot x\right)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides...

Si considerano equazioni di Ginzburg-Landau complesse del tipo $u_t-\alpha \mathrm{\Delta }u+P\left({\left|u\right|}^2\right)u=0$ in ${\mathbb{R}}^N$ dove $P$ è polinomio di grado $K$ a coefficienti complessi e $\alpha$ è un numero complesso con parte reale positiva $\mathrm{\Re }\alpha$. Nell'ipotesi che la parte reale del coefficiente del termine di grado massimo $P$ sia positiva, si dimostra l'esistenza e la regolarità di una soluzione globale nel caso $\left|\alpha \right|<C\mathrm{\Re }\alpha$, dove $C$ dipende da $K$ e $N$.

Given an open, bounded and connected set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Lipschitz boundary and volume $\left|\mathrm{\Omega }\right|$, we prove that the sequence ${\mathcal{F}}_k$ of Dirichlet functionals defined on $H^1\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$, with volume constraints $v^k$ on $m\ge 2$ fixed level-sets, and such that ${\sum }_{i=1}^m{v}_i^k<\left|\mathrm{\Omega }\right|$ for all $k$, $\mathrm{\Gamma }$-converges, as $v^k\to v$ with ${\sum }_{i=1}^m{v}_i^k=\left|\mathrm{\Omega }\right|$, to the squared total variation on $BV\left(V;{\mathbb{R}}^d\right)$, with $v$ as volume constraint on the same level-sets.

In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type $$\begin{array}{c}f(x,\eta ,\xi )\ge a\left(x\right)\frac{{\left|\xi \right|}^p}{{(1+|\eta \left|\right)}^{\alpha }}-b_1\left(x\right){\left|\eta \right|}^{{\beta }_1}-g_1\left(x\right),\\ f(x,\eta ,0)\le b_2\left(x\right){\left|\eta \right|}^{{\beta }_2}+g_2\left(x\right),\end{array}$$ where$0\le \alpha <p$, $1\le {\beta }_1<p$, $0\le {\beta }_2<p$, $\alpha +{\beta }_i\le p$, $a\left(x\right)$, $b_i\left(x\right)$, $g_i\left(x\right)$ ($i=1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}\left(a\right)$ , which assume a boundary datum $u_0\in W^{1,p}\left(a\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.