We consider nonlinear sub-elliptic systems with Dini continuous coefficients for the case 1 <m < 2 in Carnot groups and prove a C1-partial regularity result for weak solutions under the controllable growth conditions. Our method of proof for sub-elliptic systems is based on a generalization of the technique ofA-harmonic approximation. It is interesting to point out that our result is optimal in...

Abstract. Let TΩ,α (0 ≤ α < n) be the singular and fractional integrals with variable kernel Ω(x,z), and [b,TΩ,α ] be the commutator generated by TΩ,α and a Lipschitz function b. In this paper, the authors study the boundedness of [b,TΩ,α ] on the Hardy spaces, under some assumptions such as the Lr-Dini condition. Similar results and the weak type estimates at the end-point cases are also given...

When a moving (real or virtual) camera images a stationary object, the use of a rotation-minimizing directed frame (RMDF) to specify the camera orientation along its path yields the least apparent rotation of the image. The construction of such motions, using curves that possess rational RMDFs, is considered herein. In particular, the construction entails interpolation of initial/final camera p...

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f . In a reflexive Banach space X, we...

BACKGROUND Repairing DNA damage begins with its detection and is often followed by elicitation of a cellular response. In E. coli, RecA polymerizes on ssDNA produced after DNA damage and induces the SOS Response. The RecA-DNA filament is an allosteric effector of LexA auto-proteolysis. LexA is the repressor of the SOS Response. Not all RecA-DNA filaments, however, lead to an SOS Response. Certa...

In this paper, we study the obstacle problem with obstacles whose Laplacians are not necessarily Hölder continuous. We show that the free boundary at a regular point is C if the Laplacian of the obstacle is negative and Dini continuous. We also show that this condition is sharp by giving a method to construct a counter-example when we weaken the requirement on the Laplacian of the obstacle by a...

The RecA protein is a recombinase functioning in recombinational DNA repair in bacteria. RecA is regulated at many levels. The expression of the recA gene is regulated within the SOS response. The activity of the RecA protein itself is autoregulated by its own C-terminus. RecA is also regulated by the action of other proteins. To date, these include the RecF, RecO, RecR, DinI, RecX, RdgC, PsiB,...

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of ∆u − ut at the origin is Dini (in Lp average), then the origin is a Lebesgue point of continuity (still in Lp average) for D2u and ∂tu. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, ...

0 Introduction This paper will be concerned with the interior regularity theory of weak solutions to nonlinear elliptic systems of second order in divergence form of the following type: div A(x; u; Du) = 0 in : (0-1) Here is a bounded domain in IR n , u taking values in IR N , and the coeecients A(x; ; p) are in Hom(IR n ; IR N). Under natural hypotheses concerning the regularity and the growth...

We examine solutions u = PI f to ∆u − V u = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M , satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L-Sobolev space results of the form PI : B s (∂Ω) −→ Lps+1/p(Ω), and variants, when the metric tensor on M has limited regularity, described by a Hölder or a Dini-type modulus of continuity. We...