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Fanheng Xu

Publications and source records attributed to Fanheng Xu.

7 recordsLinked to original sources
arXiv2026

Interior gradient estimates for mean curvature equations on Riemannian manifolds

In this paper, we study interior gradient estimates for solutions of \[ \operatorname{div}_{M}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|_g^2}} \right) =f(x,\nabla u) \] on a complete Riemannian manifold of dimension $n\ge2$. Under a Ricci curvature lower bound and a Hessian comparison for the distance function, we derive polynomial and exponential gradient bounds under structural conditions on $f$ and its horizontal and vertical derivatives. The estimates quantify the dependence on the radius, the oscillation of the solution, and the geometric and analytic parameters. In the Euclidean case, we also improve the dependence on the radius in the gradient estimate of Xu~\cite[Theorem~1.1(d)]{Xu2026}. As applications, we obtain Liouville-type results for entire solutions under suitable curvature and growth assumptions.

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arXiv2026

Liouville theorems for stable solutions to the mean curvature equation

We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1 2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\le p_S$, and the solution is stable outside a compact set, then $u\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.

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arXiv2026

Improved Interior Gradient Estimates for the Mean Curvature Equation under Nonlinear Assumptions

In this paper, we investigate interior gradient estimates for solutions to the mean curvature equation $$ \dive \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = f(\nabla u)$$ under various nonlinear assumptions on the right-hand side. Under the weakened initial assumption $u\in C^1(B_R) \cap C^3(\{|\nabla u|>0\})$, we establish sharp gradient bounds that depend on the oscillation of the solution. These estimates are applicable to a wide class of nonlinear terms, including the specific forms arising from the elliptic regularization of the inverse mean curvature flow ($f=\varepsilon\sqrt{1+|\nabla u|^2}$ ), minimal surface equation ($f=0$) and several polynomial and logarithmic growth regimes. As applications, the gradient bounds imply uniform ellipticity of the equation away from the critical set,which allows one to apply classical elliptic regularity theory and obtain higher regularity of solutions in the noncritical region. Moreover, when the solution grows at most linearly, all cases of our results can be applied in Moser's theory to establish the affine linear rigidity of global solutions. This directly leads to the Liouville-type theorems for global solutions without requiring additional proofs.

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arXiv2026

Inner regularity and Liouville theorems for stable solutions to the mean curvature equation

Let $f\in C^1(\mathbb{R})$. We study stable solutions $u$ of the mean curvature equation \[ \operatorname{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) = -f(u) \qquad \text{in}\ Ω\subset \mathbb{R}^n. \] In the local setting we prove that $\nabla u$ satisfies inner Morrey regularity $M^{p_n}$, where \[ p_n := \left\{ \begin{array}{ll} n,\qquad & \text{if}\ 2\leq n\leq 5, \\ \frac{n}{n-4\sqrt{n-1}+4},\qquad & \text{if}\ n\geq 6, \end{array} \right. \] together with the estimate \[ \|\nabla u\|_{M^{p_n}(B_1)} \leq C \left( 1+\|\nabla u\|_{L^1(B_2)} \right). \] The exponent $p_n$ is optimal for $n\leq5$, as shown by an explicit one-dimensional example. For radial solutions we show that the symmetry center is at most a removable singularity. Globally, we establish Liouville-type theorem: any stable solution satisfying the growth condition \[ |\nabla u(x)| = \left\{ \begin{array}{lll} o(|x|^{-1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ 2\leq n\leq 10, \\ o(|x|^{-n/2+\sqrt{n-1}+1}) \ & \text{as}\ |x|\rightarrow +\infty& \text{when}\ n\geq 11, \end{array} \right. \] must be constant. In particular, no nonconstant radial stable solution exists in dimensions \(2\leq n\leq6\), which highlights a global rigidity of stable radial solutions in low dimensions and extend the classical Liouville theorem of Farina and Navarro. Several exponents appearing in our results are new for mean curvature equations, showing both similarities and differences with the corresponding theorems for semilinear equations.

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arXiv2021

Liouville's theorems to quasilinear differential inequalities involving gradient nonlinearity term on manifolds

We investigate the nonexistence and existence of nontrivial positive solutions to $Δ_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for $(p, q)$ of negative pair, even in the Euclidean space.

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arXiv2018

On existence and nonexistence of nonnegative solutions to seminlinear differential equation on Riemannian manifolds

In this paper, we give a clear cut relation between the the volume growth $V(r)$ and the existence of nonnegative solutions to parabolic semilinear problem \begin{align}\tag{*}\label{*} \left\{ \begin{array}{ll} Δu - \partial_t u + u^p = 0, \\ {u(x,0)= {u_0(x)}}, \end{array} \right. \end{align} on a large class of Riemannian manifolds. We prove that for parameter $p>1$, if \begin{align*} \int^{+\infty} \frac{t}{V(t)^{p-1}} dt = \infty \end{align*} then (\ref{*}) has no nonnegative solution. If \begin{align*} \int^{+\infty} \frac{t}{V(t)^{p-1}} dt < \infty \end{align*} then (\ref{*}) has positive solutions for small $u_0$.

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