Global existence of solutions for a nonlinearly perturbed Keller–Segel system in \mathbbR2{\mathbb{R}}^2

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.     

Conditions for the [`(?)] \bar \partial -closedness of differential forms

The $ \bar \partial $ \bar \partial -closed differential forms with smooth coefficients are studied in the closure of a bounded domain D ⊂ ℂ ⁿ . It is demonstrated that the condition of $ \bar \partial $ \bar \partial -closedness can be replaced with a weaker differential condition in the domain and differential conditions on the boundary. In particular, for the forms with harmonic coefficients the $ \bar \partial $ \bar \partial -closedness is equivalent to some boundary relations. This allows us to treat the results as conditions for the $ \bar \partial $ \bar \partial -closedness of an extension of a form from the boundary.     

Polynomial hulls and H￥H^\infty control for a hypoconvex constraint

We say that a subset of is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in , . We prove two conjectures of Helton and Marshall. Let be a smooth function on whose sublevel sets have compact hypoconvex fibers over . Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum is attained by a unique bounded analytic f which in fact is also smooth on . We also prove that if varies smoothly with respect to a parameter, so does the unique f just found. Received: 18 December 1998 / Published online: 28 June 2000     

Optimal regularity for [`(?)] b\bar \partial _b on CR manifoldson CR manifolds

In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained.     

Vojta’s conjecture on blowups of \mathbbPn{\mathbb{P}^n}, greatest common divisors, and the abc conjecture

We will prove some cases of Vojta’s conjecture on blowups of \mathbbPⁿ{\mathbb{P}^n}, using Schmidt’s subspace theorem. The results can be stated as inequalities of greatest common divisors. Moreover, from Vojta’s conjecture on one further blowup at an infinitely near point, we derive a still-open special case of the abc-conjecture.     

Local analyticity for the [`(?)]\overline \partial -Neumann problem in completely decoupled pseudoconvex domains-Neumann problem in completely decoupled pseudoconvex domains

In this paper we prove local analyticity of solutions to the -Neumann problem up to the boundary of rigid, completely decoupled pseudoconvex domains with real-analytic boundary. These are domains that are locally of the form Imw > Σ |h _k (z _k )|² with eachh _k holomorphic and vanishing only at 0. As in those earlier papers, we use purelyL ² methods and must construct a special holomorphic vector fieldM and then use carefully balanced polynomials inM to localize high powers ofT = ∂/∂t effectively, wheret = Rew.     

Localisation des résidus de Baum-Bott, courbes généralisées et K-théorie (I: feuilletages dans \Bbb C2 {\Bbb C}^2 )

Let v be a holomorphic vector field in a neighborhood of a point m ₀ in , which is a non dicritical isolated singularity. Let f = 0 be a reduced equation of the maximal separatrix V through m ₀, v _f the vector field , and the union of separatrices and pseudo-separatrices (i.e. the set of points where v and v _f are colinear). Assuming the foliations defined by v and v _f to be distinct, we prove that the Baum-Bott residue BB(c ₁ ² , v) of v at m ₀, as well as the difference PH(v) - μ between the Poincaré-Hopf index and the Milnor number of V at m ₀, are "localised" near the separatrices and pseudo-separatrices. (The particular case of generalized curves has already been studied in details in [CLS] and [Br]). We also interpret in K-theory the difference PH - μ as well as the GSV index of Gomez Mont-Seade-Verjovski, and we give a caracterisation of generalized curves in this framework, which will enable us to extend this concept in higher dimension. Received: August 25, 2000     

Boundary regularity for the [`(?)] b - Neumann\bar \partial _b - Neumann problem, Part 2

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the boundary conditions is an isomorphism. As an application, we establish sharp regularity for a canonical solution to the inhomogenous equation on the unit ball.     

Regularity of[`(?)] b\bar \partial _b , complex on nondegenerate Cauchy-Riemann manifolds, complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group G_ξ of step two near ξ. G_ξ varies with ξ. \square _b \square _b and [`(?)] <sub>b</sub>\bar \partial _b on M can be approximated by \square <sub>b</sub> \square _b and [`(?)] _b\bar \partial _b on the nilpotent Lie group G_ξ We can construct the parametrix of \square _b \square _b on M by using the parametrix of \square _b \square _b on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of \square _b \square _b and [`(?)] _b\bar \partial _b follows from the Harmonic analysis on M.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.