arXiv subjects
Publications and source records attributed to Wang Zhizhang.
In this paper, we provide examples to show that for $1 \leq k \leq n/2$, solutions to $k$-Hessian equations $S_k(D^2u)=1$ in the exterior of a strictly convex domain need not be quasiconvex, when prescribing quadratic growth at infinity. Additionally, we give a new proof for the quasiconvexity of harmonic functions in such exterior domains that decay to zero at infinity.
In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $Ω$ in $\mathbb{R}^n$ and $\tildeΩ$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $Ω$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\tildeΩ$. Our proof relies on a novel $C^2$ boundary estimate which utilizes the orthogonal invariance of hypersurfaces. Indeed, we employ some special vector fields generated by the infinitesimal rotations in $\mathbb{R}^{n+1}$ to establish the boundary $C^2$ estimates. This new approach enables us to handle the additional negative terms that arise when taking second order derivatives near the boundary.
In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex initial hypersurface satisfying certain conditions, the curvature flow exists for all time, and the normalized flow converges to a translating solution.
In this note, we show that there is some counterexample for isoperimetric inequality if the condition $(ϕ')^2-ϕ"ϕ\leq 1$ does not hold in warped product space.