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Fanheng Xu

Publications and source records attributed to Fanheng Xu.

5 recordsLinked to original sources
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.

math.AP↗
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 \[ \operatorname{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right) = -f(u) \qquad \text{in}\ \Omega \subset \mathbb{R}^n. \] 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.

math.AP↗
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.

math.AP↗
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 $u_0$.

math.AP↗