Free boundary problems in potential theory

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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. We also gave a detailed discussion about a corner with a right angle and a cusp.In addition to the contribution to our study, each of the investigators obtained his own results.Mochizuki discussed large time asymptotics of small solutions to generalized KPP equation.Ishii treated homogenization of Hamilton-Jacobi equations and Gaussian curvature flows.Kurata discussed the fundamental solution, eigenvalue asymptotics and eigenfunction of degenerate elliptic operators.Takakuwa showed a compactness theorem for harmonic maps.Hidano discussed nonlinear small data scattering of the wave equation.Hirata discussed the distribution of the return times of the dynamical systems by piecewise monotone transformations.