Study on Regularity and Singularity of a weak solution to the m-harmonic maps and the evolution

(1)m-调和映射和m-调和映射流的自由边界问题我们证明了m-调和映射流到目标流形的闭子流形上具有自由边界的光滑紧致流形的局部时间解的存在性，它满足弱意义下的m-调和映射流方程，并且在时空区域中具有H“上连续的梯度直到空间域的边界。解的最大存在时间由初始数据的m-能量估计如下。同时，解在奇异时刻(最大生存时刻)的奇异行为可以用非常数m-调和映射到目标流形来刻画。M-调和映射定义在m维球上，或具有自由边界的m维球上，它们分别称为m-调和球或m-调和圆盘。这些解正是目标流形上的极小子流形。(2)m-调和映射流的有限奇性期望奇异时的奇异集由有限多个点组成。在本文中，我们制作了一些公式，并试图证明这一猜想。然而，我们面临着一个严重的证据缺口，这是我们现在正在研究要克服的。得到了m-调和拉普拉斯算子的标度能量的单调性公式。(3)非散度化抛物型线性系统的先验估计我们证明了某些Sobolev空间中m-调和映射流的线性化抛物型系统的先验估计是成立的，并且系统强解的存在性。将解的存在性结果与Leray-Schauder不动点定理相结合，用反射法求出了具有自由边界的m-调和映射流的时间局部解。

We obtain the following results and try to publish the papers in some Journal.(1) Free boundary problem for m-harmonic maps and m-harmonic map flowWe show the existence of the local in time solution of the m-harmonic map flow into smooth compact manifold with free boundary on a closed submanifold of the target manifold, which satisfies the m-harmonic map flow equation in the weak sense and is H"older continuous with its gradient in time-space region up to the boundary of the space domain. The maximal existence time of the solution is estimated below by the m-energy of the initial datum. Also, the singular behavior of the solution at the singular time (maximal existence time) can be characterized by a non-constant m-harmonic maps into the target manifold. The m-harmonic map is defined on m-dimensional sphere, or m-dimensional ball with free boundary, and they are called m-harmonic sphere, or m-harmonic disk, respectively. These solutions are exactly minimal submanifolds in the target manifold.(2) Finite singularity of the m-harmonic map flowIt is expected that the singular set at the singular time is consist of finitely many points. In the paper, we make device of some formula and try to prove the conjecture. However, we are faced with a serious gap of the proof., which is now studied by us to be overcome. We obtain the formula which says the monotonicity of the scaled energy in the intrinsic way to the m-harmonic Laplace operator and is of its own interest.(3) A priori estimates for the linearized parabolic system of non-divergence formWe show the a priori estimates in some Sobolev space hold for the linearized parabolic system of the m-harmonic map flow and the existence of a strong solution of the system. The existence result is combined with the Leray-Schauder fixed point theorem aid the reflection method to show the local in time solution of the m-harmonic map flow with free boundary.