Inner regularity and Liouville theorems for stable solutions to the mean curvature equation
Fanheng Xu
TL;DR
This work analyzes stable solutions of the mean curvature equation $\operatorname{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right)=-f(u)$, deriving sharp interior gradient regularity and global rigidity phenomena. The authors prove an inner Morrey regularity result for the gradient, with a dimension-dependent optimal exponent $p_n$ (equal to $n$ for $2\le n\le 5$ and $p_n=\dfrac{n}{n-4\sqrt{n-1}+4}$ for $n\ge 6$), together with a bound $\|\nabla u\|_{M^{p_n}(B_1)}\le C(1+\|\nabla u\|_{L^1(B_2)})$; optimality is demonstrated in low dimensions via a one-dimensional example. Globally, a Liouville-type theorem states that stable solutions with prescribed decay of $|\nabla u|$ at infinity must be constant, and in particular there are no nonconstant radial stable solutions for $2\le n\le 6$, revealing a global rigidity phenomenon in the low-dimensional regime. A central gradient-estimation technique ties together local Morrey-type regularity and global Liouville theory, and the results draw connections with, yet also distinguish themselves from, the semilinear case studied by Cabré, Figalli, Ros-Oton, Serra, and others.
Abstract
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.