Study on Regularity and Singularity of Minimal Surfaces in Higher Dimensions and The Evolution

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We obtain the following results and prepare the papers to be published in some Journal.(1)Existence and regularity for the evolution of constant mean curvature surfaces in high dimensionIn high dimension where the domain dimension is equal to or greater than 3, the mean curvature of the parametric surfaces is given by the m-Laplace operator of the map which is the parametrization of the surface.We show that, If the initial boundary data is of small image in some sense, there exists a time-global weak solution The solution has the image of the same size as the datum, and its gradients are H"older continuous except some closed set in the domain. The size of the except set for regularity is estimated in the Hausdorff measure of some dimension.To show the existence of a weak solution, we use the variational method called discrete Morse semi-flow, which is the minimization of the family of the functionals, of which the Euler-Lagrange equations are the time-discrete equations of the Rothe ty … More pe.To have the regularity of a weak solution, we use the fundamental regularity theorem for the evolution of p-Laplace operator with lower order term of the critical growth on the gradient, which was obtained by Masashi Misawa in 2002.(2)Regularity and singularity for a singular perturbation problemWe study a singular perturbation problem in a phase transition., and in particular, we study the regularity of the interface which is the level set of the limit function, of the singular perturbation problem. To investigate the regularity and singularity of the interface of the limit function, we make device of the formula for the scaled energy, called monotonicity formula.(3)Free boundary problem for minimal surfaces in high dimensionWe study the free boundary problem for minimal surfaces in high dimension. The existence of a solution is proved by the variational method, in particular, the minimax method combined with some approximation., and the solution is nearly unstable. We also study the relation of the unstable solution with the singularity of the evolution of minimal surfaces in high dimension.. It is shown that there exists a time-global weak solution of the evolution of minimal surfaces with free boundaries in high dimension, and that the solution and its gradient is H"older continuous except finitely many times. Moreover, the singular time is characterized by the existence of a non-constant minimal surface with free boundaries.We will try to study the free boundary problem for p-harmonic maps with values into smooth compact Riemannian manifold, the evolution, of p-harmonic maps, and moreover the wave equations and wave maps into smooth compact Riemannian manifold. Less