This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation...This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR geometry.Assuming only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals-H.By introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more derivative.Therefore the seed curves are C 2 smooth.This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,respectively.In an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order.Moreover,this ODE is analyzed to study the singular set.展开更多
In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the nat...In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.展开更多
基金supported by the "Science Council" of Taiwan 11529,China (Grant No. 97-2115-M-001-016-MY3)
文摘This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR geometry.Assuming only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals-H.By introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more derivative.Therefore the seed curves are C 2 smooth.This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,respectively.In an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order.Moreover,this ODE is analyzed to study the singular set.
基金This work was supported by NSERC(Grant No.RGPIN/9319-2005)
文摘In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.
