Geodesic
Asymptotic solution of variational inequalities for a linear operator with a small parameter on the highest derivatives
Izvestiya. Mathematics , Tome 37 (1991) no. 1, pp. 97-117

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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. 
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     author = {S. A. Nazarov},
     title = {Asymptotic solution of variational inequalities for a linear operator with a small parameter on the highest derivatives},
     journal = {Izvestiya. Mathematics },
     pages = {97--117},
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     volume = {37},
     number = {1},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a4/}
}
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S. A. Nazarov. Asymptotic solution of variational inequalities for a linear operator with a small parameter on the highest derivatives. Izvestiya. Mathematics , Tome 37 (1991) no. 1, pp. 97-117. http://geodesic.mathdoc.fr/item/IM2_1991_37_1_a4/