  A high-contrast two-phase nonlinear composite material with adjacent
inclusions of $m$-convex shapes is considered for $m>2$. The mathematical
formulation consists of the insulated conductivity problem with $p$-Laplace
operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the
gradient of the solution, always blows up with respect to the distance
$\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first
establish the pointwise upper bound on the gradient possessing the singularity
of order $\varepsilon^{-\beta}$ with $\beta=(1-\alpha)/m$ for some
$\alpha\geq0$, where $\alpha=0$ if $d=2$ and $\alpha>0$ if $d\geq3$. In
particular, we give a quantitative description for the range of horizontal
length of the narrow channel in the process of establishing the gradient
estimates, which provides a clear understanding for the applied techniques and
methods. For $d\geq2$, we further construct a supersolution to sharpen the
upper bound with any $\beta>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a
subsolution is also constructed to show the almost optimality of the blow-up
rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares.
This fact reveals a novel dichotomy phenomena that the singularity of the
gradient is uniquely determined by one of the convexity parameter $m$ and the
nonlinear exponent $p$ except for the critical case of $p=m+1$ in two
dimensions.
