We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.
We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu h(x,u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \] in a bounded domain $\Omega \subset \mathbb{R}^N$, where $\la...
A. Boscaggin, F. Colasuonno, Ricardo Ziegele· 1 citation· ⚡1
A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville n...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in al...
We prove that for every integer $n\geq 2$ and $8\pi(n-1)<\rho<8\pi n$, the singular Liouville equation $\Delta u+\e^u=\rho\delta_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and...
Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U...
Dominik Schlagenhauf· 2 citations
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