arXiv subjects
Publications and source records attributed to Min-Chun Hong.
In this paper, we study an $α$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $α$-flows is a weak solution to the harmonic map flow. By an application of the $α$-flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class.
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $Σ$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $Σ$ is 1-rectifiable.
We propose a new approximation for the relaxed energy $E$ of the Dirichlet energy and prove that the minimizers of the approximating functionals converge to a minimizer $u$ of the relaxed energy, and that $u$ is partially regular without using the concept of Cartesian currents. We also use the same approximation method to study the variational problem of the relaxed energy for the Faddeev model and prove the existence of minimizers for the relaxed energy $\tilde{E}_F$ in the class of maps with Hopf degree $\pm 1$.
For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $Ω\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $λ\in [0,1]$ we introduce a $λ$-relaxed energy $\H_λ$ for the Hessian energy for maps in $W^{2,2}(Ω,S^4)$ so that each minimizer $u_λ$ of $\H_λ$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_λ$ for $λ\in [0,1)$.