Aug 2026· 2 citations· ⚡ 1 influential· 15 references
Mathematics
Abstract
We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(\Omega)\) be a weak \(p\)-harmonic function in a domain \(\Omega\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du\) is a local diffeomorphism.
Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely,...
Let $3\leq n\leq7$, $2\leq k\leq n-1$, and $m=n-k-1$. We prove that a complete, connected, noncompact spin manifold $(M^n,g)$ with scalar curvature $R_M\geq k(k-1)$ is isometric to $\mathbb{S}^k\times\mathbb{T}^m_\Lambda\times\mathbb{R}$ if it admits a smooth proper map of nonzero degree to $\mathbb{S}^k\times\mathbb{T...
We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $\Omega\subset\mathbb{R}^3$ 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 $Du(\Omega)$. We show that if...
We consider the following singular Liouville equation $-\Delta v=\lambda V(x)|x|^2e^v \quad \text{in } B_1, \quad v=0 \quad \text{on } \partial B_1, $ where $B_1\subset\mathbb R^2$ is the unit disk, $\lambda>0$ is a small parameter, and $V$ is a positive smooth function. We first prove a non-degeneracy result for the b...
In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n...
For every $0\leq\beta\leq1/2$, we construct a nonzero real-valued continuous function $f_\beta$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_\beta=f_\beta$ and $f_\beta(\sqrt{n}/[\log(e+n)]^{\beta})=0$ for all $n\geq 0$. The case $\beta=0$ settles in the negative a question raised by Radchenko and Via...
A. Bondarenko, Kristian Seip· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
