Mass and Asymptotics associated to Fractional Hardy-Schr\"odinger Operators in Critical Regimes

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schr\"odinger operator $L_{\gamma,\alpha}: = ({-}{ \Delta})^{\frac{\alpha}{2}}- \frac{\gamma}{|x|^{\alpha}}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<\alpha<2$ and $0 \le \gamma < \gamma_H(\alpha)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{\gamma,\alpha}-\lambda I)u= {\frac{u^{2^\star_\alpha(s)-1}}{|x|^s}}$ on $\Omega$, where $0\leq s <\alpha <n$ and $2^\star_\alpha(s)={\frac{2(n-s)}{n-{\alpha}}}$ is the critical fractional Sobolev exponent. We show that if $\gamma$ is below a certain threshold $\gamma_{crit}$, then such solutions exist for all $0<\lambda <\lambda_1(L_{\gamma,\alpha})$, the latter being the first eigenvalue of $L_{\gamma,\alpha}$. On the other hand, for $\gamma_{crit}<\gamma <\gamma_H(\alpha)$, we prove existence of such solutions only for those $\lambda$ in $(0, \lambda_1(L_{\gamma,\alpha}))$ for which the domain $\Omega$ has a positive {\it fractional Hardy-Schr\"odinger mass} $m_{\gamma, \lambda}(\Omega)$. This latter notion is introduced by way of an invariant of the linear equation $(L_{\gamma,\alpha}-\lambda I)u=0$ on $\Omega$

Design of a Novel Portable Flow Meter for Measurement of Average and Peak Inspiratory Flow

The maximum tolerable physical effort that workers can sustain is of significance across many industrial sectors. These limits can be determined by assessing physiological responses to maximal workloads. Respiratory response is the primary metric to determine energy expenditure in industries that use respirator masks to protect against airborne contaminants. Current studies fail to evaluate endurance under conditions that emulate employee operating environments. Values obtained in artificial laboratory settings may be poor indicators of respiratory performance in actual work environments. To eliminate such discrepancies, equipment that accurately measures peak respiratory flows in situ is needed. This study provides a solution in the form of a novel portable flow meter design that accurately measures average and peak inspiratory flow of a user wearing an M40A1 respirator mask