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

我们获得了以下结果，并尝试在一些期刊上发表论文。(1)m-调和映射流的自由边界问题我们证明了在目标流形的闭子流形上，m-调和映射流到具有自由边界的光滑紧致流形上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，并且在时空区域内梯度到空间域边界是H“older连续的.解的最大存在时间由初始数据的m-能量估计如下。此外，在奇异时间（最大存在时间）的解的奇异行为可以通过一个非常数的m-调和映射到目标流形。m-调和映射定义在m-维球面或m-维自由边界球上，它们分别称为m-调和球面或m-调和圆盘。这些解都是目标流形中的极小子流形。(2)m-调和映射流的有限奇异性我们期望奇异时刻的奇异集由100个点组成。本文给出了一些公式的设计，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。我们正在研究如何克服它。得到了m-调和拉普拉斯算子标度能量的内禀单调性公式，这是它自己的兴趣所在. (3)非散度型线性抛物方程组的先验估计我们证明了m-调和映射流线性抛物方程组的先验估计在Sobolev空间中成立，并证明了该方程组强解的存在性。将此结果与Leray-Schauder不动点定理相结合，利用反射方法给出了具有自由边界的m-调和映射流的局部时间解。

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.