New developements of the regularity theory on the solutions to the Liouville-Gelfand type problems and the related parabolic dynamics

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The aim of this research program is to solve the long open problem concerning the regularity and singularity of extremal solutions to the Liouville-Gelfand type nonlinear eigenvalue problems in a unified way, with no assumptions of the domains or the symmetry of solutions, by combining ideas from the geometric variational problems and the techniques of nonlnear analysis.The research outputs by the research group in the period of the program, 2012-2013, are the following: 20 published papers (including accepted papers), 7 talks at Math. Soc. Japan, 21 talks by the reseach representative at seminars, including abroad.