Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.     

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the p$p$-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.     

Fat sets and pointwise boundary estimates forp-harmonic functions in metric spaces

We extend a result of John Lewis [L] by showing that if a doubling metric measure space supports a (1,q ₀)-Poincaré inequality for some 1<q ₀<p, then every uniformlyp-fat set is uniformlyq-fat for someq<p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation ofp-harmonic functions andp-energy minimizers near a boundary point.     

Low-Level Quantitation of Formaldehyde in Drug Substance by HPLC–UV

Formaldehyde has been highlighted as potential genotoxic impurity (GTI). Trace-level quantification of GTIs in drug substances requires sensitive, precise and accurate analytical methodologies for their estimation in drug substances and control. Analysis and estimation of formaldehyde is very challenging due to its properties namely volatility, high polarity, low molecular weight and over and above the absence of chromophore. This article presents a validated HPLC–UV method which is sensitive to quantification of formaldehyde in active pharmaceutical ingredient. As formaldehyde does not possess chromophore, the developed HPLC method involves derivatization with 2,4-dinitrophenylhydrazine. Using this method, the detection and quantitation limits achieved are 0.5 and 1.5 ppm, respectively. The calibration curve of formaldehyde was linear over the concentration range of 1.5–20 ppm. The method was found to be sensitive, precise and accurate and the proposed method has been successfully applied to estimate formaldehyde content in scale-up batches of bulk drug.

     

Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces

In the setting of metric measure spaces equipped with a doubling measure supporting a weak p-Poincaré inequality with 1?p<∞, we show that any uniform domain Ω is an extension domain for the Newtonian space N1,p(Ω) and that Ω, together with the metric and the measure inherited from X, supports a weak p-Poincaré inequality. For p>1, we obtain a near characterization of N1,p-extension domains with local estimates for the extension operator.     

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L ^n,∞. We also obtain the a priori global and local estimates for the L ^n,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.     

SNi′ displacements with main group organometallics

1. Download : Download full-size image