Asymptotic solution of a variational inequality modelling a friction

S. A. Nazarov

Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 337-369

Résumé

The problem of minimizing the nondifferentiable functional $$ \mu^2(\nabla u,\nabla u)_\Omega\times (u,u)_\Omega -2(f,u)_\Omega+\gamma(|u|,g)_{\partial\Omega} $$ is considered. An asymptotic solution of the corresponding variational inequality is constructed and justified under the assumption that $\mu$ or $\gamma$ is a small parameter. Also, formal asymptotic representations are obtained for singular surfaces which characterize a change in the type of boundary conditions. For $\mu\to 0$ a modification of the Vishik–Lyusternik method is used, and exponential boundary layers arise. If $\gamma\to 0$, then the boundary layer has only power growth; the principal term of the asymptotic expansion of the solution of the problem in a multidimensional region $\Omega$ and the complete asymptotic expansion for the case $\Omega\subset\mathbf R^2$ are obtained.