Let $\mathrm{\Omega }$ be a bounded open subset of $R^n$, $n>4$, of class $C^2$ . Let $u\in H^2\left(\mathrm{\Omega }\right)$ a solution of elliptic non linear non variational system $$a\left(x,u,Du,H\left(u\right)\right)=b\left(x,u,Du\right)$$ where $a\left(x,u,\mu ,\xi \right)$ and $b\left(x,u,\mu \right)$ are vectors in $R^N$, $N\ge 1$, measurable in $x$, continuous in $\left(u,\mu ,\xi \right)$ and $\left(u,\mu \right)$ respectively. Here, we demonstrate that if $b\left(x,u,\mu \right)$ has limit controlled growth, if $a\left(x,u,\mu ,\xi \right)$ is of class $C^1$ in $\xi$ and satisfies the Campanato condition $\left(A\right)$ and, together with $\frac{\partial a}{\partial \xi }$, certain continuity assumptions, then the vector $Du$ is partially Hölder continuous for every exponent $\alpha <1-\frac{n}{p}$.

Questo articolo considera una successione di equazioni differenziali a derivate parziali non lineari in forma di divergenza del tipo $$-\text{div}\left(Q^{ϵ}G\left(x,N^{ϵ}\nabla u\right)\right)=f^{ϵ},$$ in un dominio limitato $\mathrm{\Omega }$ dello spazio $n$-dimensionale; $Q^{ϵ}=Q^{ϵ}\left(x\right)$ e $N^{ϵ}=N^{ϵ}\left(x\right)$ sono matrici con coefficenti limitati, $N^{ϵ}$ e è invertibile e la sua matrice inversa $R^{ϵ}$ ha anche coefficenti limitati. La non linearità è dovuta alla funzione $G=G\left(x,\xi \right)$; la condizione di crescita, la monotonicità e le ipotesi di coercitività sono modellate sul $p$-Laplaciano, $1<p<\mathrm{\infty }$, ed assicurano l'esistenza di...

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 $u_0\in W^{1,p}\left(a\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

Let $H_n$ be the Heisenberg group of dimension $2n+1$. Let ${\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n$ be the partial sub-Laplacians on $H_n$ and $T$ the central element of the Lie algebra of $H_n$. We prove that the kernel of the operator $m\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n,-iT\right)$ is in the Schwartz space $S\left(H_n\right)$ if $m\in S\left({\mathbb{R}}^{n+1}\right)$. We prove also that the kernel of the operator $h\left({\mathcal{L}}_1,\mathrm{\dots },{\mathcal{L}}_n\right)$ is in $S\left(H_n\right)$ if $h\in S\left({\mathbb{R}}^n\right)$ and that the kernel of the operator $g\left(\mathcal{L},-iT\right)$ is in $S\left(H_n\right)$ if $g\in S\left({\mathbb{R}}^2\right)$. Here $\mathcal{L}={\mathcal{L}}_1+\mathrm{\dots }+{\mathcal{L}}_n$ is the Kohn-Laplacian on $H_n$.