arXiv subjects
Publications and source records attributed to Rongli Huang.
We establish the existence of smooth solutions to the symmetric maximal surface equation with degenerate boundary conditions. Moreover, we prove that these solutions maximize the associated area functionals. This result serves as the Lorentzian analogue of minimal graphs in hyperbolic spaces together with their associated area minimizing problem.
We establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation. Since the curvature matrix depends on both $Du$ and $D^2u$, a phase subsolution alone does not provide the full linearized separation needed for the mixed derivative estimate. We introduce a mixed recession compatibility condition imposed only on doubly degenerate level jets. It yields a uniform mixed derivative bound and is sharp within the class of fixed smooth zero-order barriers considered here. For the double-normal derivative, an exact complex Schur-complement identity gives \[ u_{\nu\nu}=\beta+\alpha\cot\delta, \qquad 1\leq\alpha\leq C, \qquad |\beta|\leq C, \] where $\alpha$ and $\beta$ are explicit Schur-complement coefficients, $\delta$ is the actual boundary limiting-phase gap, and $C$ depends only on uniform bounds for the gradient and the mixed boundary derivatives. Thus curvature blows up if and only if this gap collapses, with optimal rate $\delta^{-1}$. Smooth radial solutions attain the rate, while a rank-loss model shows that a strict lower subsolution need not force strict convexity.
In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $Ω$ and $\tildeΩ$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of mean curvature type equations.
We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.
In this paper, we consider the existence of constant mean curvature hypersurfaces with prescribed gradient image. Let $Ω$ and $\tildeΩ$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of constant mean curvature equations.
In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation \eqref{11} has a smooth solution $u(x,t)$ for three corresponding nonlinear equations between the Monge-Amp$\grave{e}$re type equation($τ=0$) and the special Lagrangian parabolic equation($τ=\fracπ{2}$). Furthermore, we get the bound of $D^lu$, $l=\{3,4,5,\cdots\}$ for $τ=\fracπ{4}$ and the decay estimates of the higher order derivatives when $0<τ<\fracπ{4}$ and $\fracπ{4}<τ<\fracπ{2}$. We also prove that $u(x,t)$ converges to smooth self-expanding solutions of \eqref{12}.
This is a sequel to [2] and [3], which study the second boundary value problems for mean curvature flow. Consequently, we construct the translating solitons with prescribed Gauss image in Minkowski space.
In this paper, we study a class of special Lagrangian curvature potential equations and obtain the existence of smooth solutions for Dirichlet problem. The existence result is based on a priori estimates of global $C^{0}$, $C^{1}$ and $C^{2}$ norms of solutions under the assumption of existence of a subsolution.
This is a sequel to [1] and [2], which study the second boundary problem for special Lagrangian curvature potential equation. As consequences, we obtain the existence and uniqueness of the smooth uniformly convex solution by the method of continuity through carrying out a-priori estimate on the solutions.
Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle-Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.
We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.
The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.