Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.39276

Interior curvature estimates for graphical curvature quotient equations

Fei Han·Yan Ma
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove interior curvature estimates for admissible graphical solutions of curvature quotient equations in the range $3\leq k<n$ on the full Gårding cone $Γ_k$. No convexity, semiconvexity, or $Γ_{k+1}$-admissibility assumption is required. The proof combines a new quantitative compression inequality, a two-surface comparison and doubling argument, and a Pogorelov estimate.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fei Han, Yan Ma. 2026-09-30. Interior curvature estimates for graphical curvature quotient equations. https://arxiv.org/abs/2609.39276

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Global Well-Posedness of the Vacuum Free Boundary Problem for the Degenerate Compressible Navier-Stokes Equations With Large Data of Spherical Symmetry

The study of global-in-time dynamics of vacuum is crucial for understanding viscous flows. In particular, physical vacuum, which ensures a finite, nontrivial pressure-induced acceleration in the normal direction on the moving boundary, naturally arises in the motion of gaseous stars and shallow water. The corresponding large-data problems for spherically symmetric flows remain open, due to the combined difficulties of coordinate singularity at the origin and degeneracy on the moving boundary. In this paper, we analyze the free boundary problem for the barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients (as in the shallow water equations) in two and three spatial dimensions. For a general class of spherically symmetric initial densities: $ρ_0^β\in H^3$ with $β\in (\frac{1}{3},γ-1]$ ($γ$: adiabatic exponent), vanishing on the moving boundary in the form of a distance function, we establish the global existence of the unique classical solution with large data. Moreover, our solution is classical up to the moving boundary, and the physical vacuum is allowed for the data under consideration here when $β=γ-1$. The key ingredient of our analysis is to obtain the uniform control over the first-order spatial derivatives of the flow map $η(t,r)$ in spherically symmetric Lagrangian coordinates, which is achieved by some novel density-weighted estimates and flow-map-weighted estimates for the solutions.

math.AP↗
arXiv2026

Global solutions to the compressible Navier--Stokes--Poisson equations in the half-space and exterior domains

We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order $-s$, with $\frac12\leq s<1$, yield the rate $(1+t)^{-s/2}$. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for $0<s<\frac14$. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.

math.AP↗
arXiv2026

Nonlocal Equations on Domains and Manifolds Without Symmetries

In this work, we establish existence and regularity results for solutions to nonlinear nonlocal equations of the form $f(Δ)u = U(x,u)$ and $f(Δ_{g})u = U(x,u)$ in Euclidean and Riemannian settings, respectively. We prove existence results across three distinct geometric scenarios: bounded domains in $\mathbb{R}^n$, perturbed equations on compact Riemannian manifolds, and noncompact Cartan-Hadamard manifolds under suitable geometric curvature conditions.

math.AP↗