Boundary regularity for elliptic systems under a natural growth condition

We consider weak solutions $u \in u_0 + W^{1,2}_0(\Omega,R^N) \cap L^{\infty}(\Omega,R^N)$ of second order nonlinear elliptic systems of the type $- div a (\cdot, u, Du) = b(\cdot,u,Du)$ in $\Omega$ with an inhomogeneity satisfying a natural growth condition. In dimensions $n \in \{2,3,4\}$ we show that $\mathcal{H}^{n-1}$-almost every boundary point is a regular point for $Du$, provided that the boundary data and the coefficients are sufficiently smooth.Comment: revised version, accepted for publication in Ann. Mat. Pura App

Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problems

We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uniformly and uniformly elliptic problems

Using Training to Increase Standardization and Collaboration Across Teams

This capstone paper analyzes the effectiveness of a company’s current weekly training and its relationship with communication and dispersed teams that work in geographically separated environments. This paper examines a government contracting project and how the inconsistency of the deliverables produced by team members necessitate a need for training improvements. Research from Blayney, Kalyuga, and Sweller, (2015) and Poell, Krogt, Vermulst, Harris, and Simons (2006) on the role of workplace trainers and cognitive load theory support the training improvements presented in this paper. The proposed changes include the introduction of a standardized method of preparing for training and resources to assist trainers in the development of effective training. Additional tools provided are feedback and evaluation forms to be used to gather information on training needs

The Current State of Professional Development in Appalachia

Tied to the current federal legislation, the No Child Left Behind Act (NCLB), and all the national influences on American Education which have come before is the need for teachers to receive high-quality professional development. Approximately 5,500 National Board Certified Teachers (NBCTs) work in the 13 states that lie along the Appalachian Mountain Region of the United States. To complete this study, a stratified random sample was performed with 650 NBCTs contacted by mail. They were asked to complete and return The Beck\u27s Professional Development Inventory (BPDI). In order to analyze the data and to determine if statistical significance was achieved, Chi-square and Kruskal-Wallis nonparametric tests were performed. The goal of this study was to determine if the NBCTs working in Appalachia perceive receiving high-quality professional development activities as defined by Title IX, Section 9101 (34)A of the No Child Left Behind legislation. The Chi-square statistic confirmed the participants\u27 distribution of frequencies did not occur by chance and that their perceptions did have a pattern of preference. Statistical significance was attained at p \u3c .05 with a probability level of .000. The Chi-square frequencies that resulted from participant responses revealed a variety of teacher perceptions in the occurrence of the 18 activities. Of those persons responding, 72% perceived their professional development activities were aligned with and directly related to state academic content standards, student academic achievement standards, and assessments. While 56% perceived professional development activities as having improved and increased their knowledge of the academic subjects they teach only 42% perceived professional development activities as high-quality, sustained, intensive, and classroom focused in order to have a positive and lasting impact on classroom instruction and the teacher\u27s performance in the classroom. Respondents perceived professional development activities as being developed with extensive participation of teachers and providing the opportunity to improve classroom management skills occurring merely 36% of the time

Optimal regularity of isoperimetric sets with H\"older densities

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild ($\alpha$-)H\"older regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class $C^{1,\frac{\alpha}{2-\alpha}}$. This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends $\alpha$-H\"older continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint)

Exponential convergence to equilibrium for coupled systems of nonlinear degenerate drift diffusion equations

We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter ε≥0. The nonlinearities and potentials are chosen such that in the decoupled system for ε=0, the evolution is metrically contractive, with a global rate Λ>0. The coupling is a singular perturbation in the sense that for any ε>0, contractivity of the system is lost. Our main result is that for all sufficiently small ε>0, the global attraction to a unique steady state persists, with an exponential rate Λε=Λ−Kε. The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities

Estimating Watershed Residence Times in Artificially-Drained Landscapes and Relation to Nutrient Concentrations

Nutrient runoff from agricultural lands feeds harmful algae blooms that create a variety of problems in freshwater ecosystems. In order to reduce the effects of this nutrient runoff, Best Management Practices (BMPs) are being put in place in agricultural lands. Most of these BMPs focus on slowing down the flow of water through the watershed to give nutrient concentrations time to deplete before the water flows to the stream or river. However, the effectiveness of these BMPs are highly unknown and the process of monitoring nutrient runoff is often complex and costly. The data in this study consists of 7 years of existing water stable isotope data and 9 years of nutrient concentrations collected by volunteers of the nonprofit Wabash River Enhancement Corporation (WREC). Samples are taken twice a year (spring and fall) across a fairly large area draining into the Wabash river. We use stable isotopes, deuterium and oxygen-18 as a proxy for residence times and correlate these residence times with land use and nutrient concentrations