Changes of configuration in free boundary problems

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We studied a flow produced by injection of fluid into the narrow gap between two parallel planes, which is called a Hele-Shaw flow, and discussed the shape of the flow for immediately after the initial time. This is a typical free or moving boundary problem described by elliptic equations. We applied potential theoretic methods to the problem and succeeded in obtaining more accurate descriptions of the flow than before. We treated the case that the initial domain has a corner on the boundary. If the interior angle is less than a right angle, then the corner persists for some time with the same interior angle, whereas if the angle is greater than a right angle, then the corner disappears immediately after the initial time. The other critical case is the case that the interior angle equals to 2π. We gave a detailed discussion in this case and obtained good conditions for the corner to be a laminar-flow point.In addition to the contribution to our study, each of the investigators obtained his own results.Okada studied numerical harmonic analysis and obtained estimates on approximations of functions.Kurata discussed nonnegative solutions of nonlinear elliptic equations and obtained the existence theorem for certain unbounded domains.Takakuwa showed the uniqueness theorem for p-harmonic maps by applying the generalized Pohozaev identity.Ishii treated homogenization of Hamilton-Jacobi equations and discussed the rate of convergence of the homogenization.Mochizuki discussed large time asymptotics of eigenfunctions and spectral representation.