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

Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions