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 $\alpha <1-\frac{n}{p}$.

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

The aim of this work is to prove a chain rule and an $L^1$-lower semicontinuity theorems for integral functional defined on $BV(\mathrm{\Omega })$. Moreover we apply this result in order to obtain new relaxation and $\mathrm{\Gamma }$-convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$.

$\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, of class $C^2$ and $T>0$. In the cylinder $Q=\mathrm{\Omega }\times \left(0,T\right)$ we consider non variational basic operator $a\left(H\left(u\right)\right)-\partial u/\partial t$ where $a\left(\xi \right)$ is a vector in ${\mathbb{R}}^N$, $N\ge 1$, which is continuous in $\xi$ and satisfies the condition (A). It is shown that $\mathrm{\forall }f\in L^2\left(Q\right)$ the Cauchy-Dirichlet problem $u\in W_0^{2,1}\left(Q\right)$, $a\left(H\left(u\right)\right)-\partial u/\partial t=f$ in $Q$, has a unique solution. It is further shown that if $u\in W_0^{2,1}\left(Q\right)$ is a solution of the basic system $a\left(H\left(u\right)\right)-\partial u/\partial t=0$ in $Q$, then $H\left(u\right)$ and $\partial u/\partial t$ belong to $H_{loc}^1\left(Q\right)$. From this the Hölder continuity in $Q$ of the vectors $u$ and $Du$ are deduced respectively when $n\le 4$ and...

Let us consider a Young's function $\mathrm{\Phi }:{ℝ}^{+}\to {ℝ}^{+}$ satisfying the ${\mathrm{\Delta }}_2$ condition together with its complementary function $\mathrm{\Psi }$, and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: $B$ the unit ball of ${ℝ}^n$. In this paper we give a necessary and sufficient condition for the $L^{\varphi }$-solvability of the problem, where $L^{\varphi }$ is the Orlicz Space generated by the function $\mathrm{\Phi }$. This means solvability for $f\in L^{\mathrm{\Phi }}$ in the sense of [5], [8], where the case $\mathrm{\Phi }(t)=t^p$ is treated. ...

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus \{0\}$, $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every...