发表机构
Harbin Normal University（哈尔滨师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明梯度依赖椭圆方程的Lewy型非退化性：局部同胚梯度映射导致Hessian行列式非零，且凸环上无临界点、水平集严格凸的容量解其Hessian惯性为(1,n-1)，从而梯度为全局微分同胚，适用于p-Laplace和极小曲面方程。
AI 中文摘要
我们证明了两个互补的Lewy型定理，针对系数仅依赖于梯度的椭圆方程。首先，设$\Omega\subset\mathbb{R}^3$，令$u$满足\\[ a^{ij}(Du)u_{ij}=0, \\] 其中$a$是包含$Du(\Omega)$的开集上的光滑、对称、正定矩阵场。我们证明，如果梯度映射$Du$是局部同胚，则$\det D^2u$永不消失；因此$Du$是局部$C^\infty$-微分同胚。其次，在任意维数中，我们考虑凸环上的容量解。如果解没有临界点且其水平超曲面严格凸，则仅椭圆性就迫使Hessian具有一个正特征值和$n-1$个负特征值，即惯性指数为$(1,n-1)$。因此，梯度是到梯度空间中径向参数化环上的全局微分同胚。两个结果都适用于$p$-Laplace方程和极小曲面方程。对于全局极小曲面结果，假设光滑解的存在性。经典凸环结果提供了全局推论所需的非临界性和严格水平集凸性。由于两个系数矩阵在相关梯度范围内是实解析的，相应的局部和全局梯度微分同胚是实解析的。
英文摘要
We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $Ω\subset\mathbb{R}^3$ and let $u$ solve \[ a^{ij}(Du)u_{ij}=0, \] where $a$ is a smooth, symmetric, positive definite matrix field on an open set containing $Du(Ω)$. We show that if the gradient map $Du$ is a local homeomorphism, then $\det D^2u$ never vanishes; hence $Du$ is a local $C^\infty$-diffeomorphism. Second, in every dimension, we consider capacitary solutions on convex rings. If the solution has no critical points and its level hypersurfaces are strictly convex, then ellipticity alone forces the Hessian to have one positive and $n-1$ negative eigenvalues, that is, inertia $(1,n-1)$. Consequently, the gradient is a global diffeomorphism onto a radially parametrized ring in gradient space. Both results apply to the $p$-Laplace and minimal surface equations. For the global minimal-surface result, existence of a smooth solution is assumed. Classical convex-ring results supply the noncriticality and strict level-set convexity needed in the global corollaries. Since the two coefficient matrices are real analytic on the relevant gradient ranges, the corresponding local and global gradient diffeomorphisms are real analytic.
