Mathematical analysis and numerical simulation of a strongly nonlinear singular model

Abstract

In this paper, we are interested in the one-dimensional singular optimization problem with constraints:
\begin{equation*}
\text{Min}
\left\lbrace
\mathcal{J}(v) = \frac{1}{p} \displaystyle \int_{-1}^{1} \left| v_{x} \right|^p+ \ \frac{1}{\gamma-1} \displaystyle \int_{-1}^{1} v^{1-\gamma},\\
\\
v(\pm 1)=0 \ \text{and} \ v(0)=d
\right\rbrace,
\end{equation*}
where $1<p<\infty$, $1 < \gamma < \frac{2p-1}{p-1}$ and $d>0$.

In the first part of the paper, we show the existence of a critical value $d^{*}>0$ such that if $d \leq d^{*}$, $\mathcal{J}$ admits a minimum in a carefully chosen closed convex set of $W^{1,p}_{0}(-1,1)$.
The second part of the paper is dedicated to numerical simulations. We elaborate a numerical algorithm that transforms our constrained optimization problem into the solution of a system of ordinary differential equations. Illustrative examples are given to verify the efficiency and accuracy of the proposed numerical method to test the relevance of the proposed approach. We point out that the numerical results obtained are in good agreement with the physical phenomenon of pleated graphene in the particular case p=4 and $\gamma=9/5$ [12].