Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

Sagun Chanillo; Juan J. Manfredi

Home/Archive/arXiv

arXiv · 0704.2833

Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

Sagun Chanillo·Juan J. Manfredi

View original source ↗Read source document ↗Export metadata ↓

Abstract

We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence form with bounded measurable coefficients. The method of proof is through a Bochner technique. The Heisenberg group is seen to be en extremal manifold for our inequality in the class of manifolds whose Ricci curvature is non-negative.

Explore related subjects

math.AP math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sagun Chanillo, Juan J. Manfredi. 2007-04-21. Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds. https://arxiv.org/abs/0704.2833

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

KEEP EXPLORING

Related papers

arXiv2026

Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank

Interacting particle systems are known for their ability to generate large-scale self-organized structures from simple local interaction rules between each agent and its neighbors. In addition to studying their emergent behavior, a main focus of the mathematical community has been concentrated on deriving their large-population limit. In particular, the mean-field limit consists of describing the limit system by its population density in the product space of positions and labels. The strategy to derive such limits is often based on a careful combination of methods ranging from analysis of PDEs and stochastic analysis, to kinetic equations and graph theory. In this article, we focus on a generalization of multi-agent systems that includes higher-order interactions, which has largely captured the attention of the applied community in the last years. In such models, interactions between individuals are no longer assumed to be binary (i.e. between a pair of particles). Instead, individuals are allowed to interact by groups so that a full group jointly generates a non-linear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly non uniform and of unbounded rank. For the first time in the literature, we show that when the interaction kernels are regular enough, then the mean-field limit is determined by a limiting Vlasov-type equation, where the hypergraph limit is encoded by a so-called UR-hypergraphon (unbounded-rank hypergraphon), and where the resulting mean-field force admits infinitely-many orders of interactions.

math.AP↗

arXiv2026

Gamma Convergence and Smooth Exact-segregation Approximation for Partially Segregated Systems

We study three-component Dirichlet energies subject to the partial segregation constraint $u_1u_2u_3=0$. Our main result is a smooth approximation theorem on bounded Lipschitz domains in arbitrary dimension. For compatible nonnegative Lipschitz boundary data, every admissible $H^1$ configuration can be approximated strongly in $H^1$ by globally Lipschitz maps that are smooth in the interior and preserve both the prescribed boundary trace and the segregation constraint exactly. The construction combines a Lipschitz retraction, variable-radius mollification, and smooth truncation preserving existing zeros, without assumptions on interfaces or junctions. Under the weaker assumption of compatible nonnegative $H^{1/2}$ boundary data, we establish existence of minimizers and $Γ$-convergence in $L^2$ for the energies with product penalization $\varepsilon^{-1}\int_Ω(u_1u_2u_3)^2\,dx$. We also identify the exact $L^2$ distance to the constrained class and construct segregated corrections with controlled Dirichlet energy. The resulting correction estimate has order $\varepsilon^{1/6}$ for bounded-energy configurations; this exponent is sharp in general and improves to $o(\varepsilon^{1/6})$ for almost-minimizers.

math.AP↗

arXiv2026

Capillary John ellipsoid theorem with applications to capillary curvature problems

In this paper, we apply a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_{+}}$. This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining $C^{0}$ estimates for solutions to some capillary curvature problems (including the capillary $L_{p}$ Christoffel-Minkowski problem and the capillary $L_{p}$ curvature problem), based on the corresponding gradient estimates. As an application, we study the capillary $L_{p}$ dual Minkowski problem. A gradient estimate, together with the non-collapsing estimate and a $C^2$ estimate, yields existence for every $1 1$, with uniqueness up to dilation. We further obtain existence and uniqueness for $p>q$ without an upper restriction on $q$.

math.AP↗