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Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation.

We define a family of functionals, called \begin{document}$ p $\end{document} -oscillation functionals, that can be interpreted as discrete versions of the classical total variation functional for \begin{document}$ p = 1 $\end{document} and of the \begin{document}$ p $\end{document} -Dirichlet functionals for \begin{document}$ p>1 $\end{document} . We introduce the notion of minimizers and prove existence of solutions to the Dirichlet problem. Finally we provide a description of Class A minimizers (i.e. minimizers under compact perturbations) in dimension \begin{document}$ 1 $\end{document} .

We obtain some Poincar\'{e} type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$., Comment: Minor changes