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We study small perturbations of the Dirichlet problems for second orderelliptic equations that degenerate on the boundary. The limit of the solution,as the perturbation tends to zero, is calculated. The result is based on acertain asymptotic self-similarity near the boundary, which holds in thegeneric case. In the last section, we briefly consider the stabilization ofsolutions to the corresponding parabolic equations with a small parameter.Metastability effects arise in this case: the asymptotics of the solutiondepends on the time scale. Initial-boundary value problem with the Neumannboundary condition is discussed in the last section as well.

Asymptotics in the Dirichlet Problem for Second Order Elliptic Equations with Degeneration on the Boundary