Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions

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.