Asymptotic solution of variational inequalities for a linear operator with a small parameter on the highest derivatives

Full asymptotic expansions are found and justified for solutions of problems with smooth obstructions on the boundary $\partial\Omega$ and in the domain $\Omega\subset\mathbf R^n$ for the operator $-\varepsilon^2\Delta^2+1$ with a small parameter $\varepsilon$ on the highest derivatives. In the construction of the asymptotics of solutions one formally computes an asymptotic expansion of the equation that yields a singular submanifold (for example, of a surface where the type of the boundary conditions changes). Near such surfaces there occur additional boundary layers, which are determined by solving both ordinary and partial differential equations.