Singular quasilinear convective elliptic systems in ℝ N

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

Prefrontal Activation is Associated With Gait Quality During an Attentional Task in Older Adults

Prefrontal cortical activation varies by walking task and is a marker of attentional demand. We compared prefrontal activation by functional near-infrared spectroscopy (fNIRS) to accelerometry-derived gait quality. We hypothesized greater activation with lower gait quality (greater step-time coefficient-of-variation, decreased cadence, smoothness, regularity, and signal variability). Participants (n=114; age 74.4±6.0 years, 59.6% female) were independently ambulating individuals >64 years. Attentional (reciting every-other alphabet letter) and physical (uneven surface) challenges mimicked community mobility and provided four 15m walking conditions: even, uneven, ABC-even, and ABC-uneven. fNIRS data were referenced to quiet standing and averaged within left and right hemispheres. Gait metrics from a tri-axial accelerometer at the lower-back included cadence (steps/min), step-time coefficient-of-variation, signal variability (standard deviation), smoothness (harmonic ratio), and regularity (entropy). Associations between fNIRS and gait were quantified using Pearson correlations (α=0.05). Results were consistent across hemispheres, gait axes, and robust to adjustment for age and gait speed; we report unadjusted coefficients for left hemisphere and anterior-posterior gait direction. Greater prefrontal activation was associated with slower cadence (r=-0.220, p=0.019), lower signal variability (r=-0.228, p=0.015), and reduced smoothness (r=-0.194, p=0.039) during ABC-even. No relation was observed for step-time coefficient-of-variation or regularity. Results were similar for the ABC-uneven condition, except there was no association with gait smoothness but was with step-time coefficient-of-variation (r=0.25, p=0.007). Prefrontal activation was not correlated to gait quality in non-ABC conditions. Our findings support our hypothesis only during the ABC challenge, suggesting that older adults may rely on prefrontal activation to complete attentional but not physical challenges during gait.

Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.