Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions
| Autor: | Chen, Qionglei, Zhao, Zhiwen |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: exposition improved, a section is added for further discussions and remarks in the end of the paper |
| Databáze: | arXiv |
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