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Let $\Omega$ be a bounded open subset of ${ℝ}^n$, let $X=(x,t)$ be a point of ${ℝ}^n\times {ℝ}^N$. In the cylinder $Q=\Omega \times (-T,0)$, $T>0$, we deduce the local differentiability result $$u\in L^2(-a,0,H^2(B\left(\sigma \right),{ℝ}^N))\cap H^1(-a,0,L^2(B\left(\sigma \right),{ℝ}^N))$$ for the solutions $u$ of the class $L^q(-T,0,H^{1,q}(\Omega ,{ℝ}^N))\cap C^{0,\lambda }(\overline{Q},{ℝ}^N)$ ($0<\lambda <1$, $N$ integer $\ge 1$) of the nonlinear parabolic system $$-\sum _{i=1}^n{D}_i{a}^i(X,u,Du)+{\displaystyle \frac{\partial u}{\partial t}}=B^0(X,u,Du)$$ with quadratic growth and nonlinearity $q\ge 2$. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1,q}\cap C^{0,\lambda }$.

Let $\Omega$ be a bounded open subset of ${ℝ}^n$, $n>2$. In $\Omega$ we deduce the global differentiability result $$u\in H^2(\Omega ,{ℝ}^N)$$ for the solutions $u\in H^1(\Omega ,{ℝ}^n)$ of the Dirichlet problem $$u-g\in H_0^1(\Omega ,{ℝ}^N),-\sum _i{D}_i{a}^i(x,u,Du)=B_0(x,u,Du)$$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

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