arXiv subjects
Publications and source records attributed to Xi-Nan Ma.
In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the $k$-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also the solutions of the classical Neumann problem converge to a translating solution.
We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case.
For the Monge-Ampère equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation.
In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.
In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give a affirmative answer to a conjecture of N. Trudinger in 1986.
In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR) Yamabe problem, Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg group $\mathbb H^n$ by using the computer in [11]. They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae. With the help of dimensional conservation and invariant tensors, we can answer the above question. For a class of subcritical exponent subelliptic equations on the CR manifold, several new types of differential identities are found. Then we use those identities to get the rigidity result, where rigidity means that subelliptic equations have no other solution than some constant at least when parameters are in a certain range. The rigidity result also deduces the sharp Folland-Stein inequality on closed CR manifolds.
In this paper, we consider the homogeneous complex k-Hessian equation in $Ω\backslash\{0\}$. We prove the existence and uniqueness of the $C^{1,α}$ solution by constructing approximating solutions. The key point for us is to construct the subsolution for approximating problem and establish uniform gradient estimates and complex Hessian estimates which is independent of the approximation.
In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\inΩ$ where $Ω$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $Ω\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hessian equation. Then we derive the pointwise $C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$-convex starshaped closed hypersurface in $\mathbb R^n$ with $\frac{n}{2}\le k<n$.
We use the integral by parts to get a Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere, this $p$-Laplace type equation arises from the study of asymptotic behavior near the origin for the semi-linear $p$-Laplace equation on the puncture ball $B_1(o) \subset R^n$. This gives a positive answer to L. Véron's question in a paper \cite{Veron92} and his book \cite{Veron} at page 440.
In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminusΩ$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.
In this paper, we use the maximum principle to get the gradient estimate for the solutions of the prescribed mean curvature equation with Neumann boundary value problem, which gives a positive answer for the question raised by Lieberman \cite{Lieb13} in page 360. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. Moreover we can get a new proof of the gradient estimates for the mean curvature equation with prescribed contact angle boundary value problem.
This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 2.2.