Boundary clustered layer positive solutions for an elliptic Neumann problem with large exponent

Let $\mathcal{D}$ be a smooth bounded domain in $\mathbb{R}^N$ with $N\geq3$, we study the existence and profile of positive solutions for the following elliptic Neumann problem $$\begin{cases}-\Delta \upsilon+\upsilon=\upsilon^p,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\\[1mm] \frac{\partial \upsilon}{\partial\nu}=0\qquad\textrm{on}\ \partial\mathcal{D}, \end{cases}$$ where $p>1$ is a large exponent and $\nu$ denotes the outer unit normal vector to the boundary $\partial\mathcal{D}$. For suitable domains $\mathcal{D}$, by a constructive way we prove that, for any integers $l$, $m$ with $0\leq l\leq m$ and $m\geq1$, if $p$ is large enough, such a problem has a family of positive solutions with $l$ interior layers and $m-l$ boundary layers which concentrate along $m$ distinct $(N-2)$-dimensional minimal submanifolds of $\partial\mathcal{D}$, or collapse to the same $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $p\rightarrow+\infty$

Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-H\'{e}non weight and large exponent

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$\begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, \Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partial\Omega, \end{cases}$$ where $\nu$ denotes the outer unit normal vector to $\partial\Omega$, $q\in\overline{\Omega}$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions. We show that if $q\in\Omega$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,\nu(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overline{\Omega}$ is an isolated local maximum point of $a(x)$ or not.Comment: arXiv admin note: text overlap with arXiv:1904.0293

Special Power Electronics Converters and Machine Drives with Wide Band-Gap Devices

Power electronic converters play a key role in power generation, storage, and consumption. The major portion of power losses in the converters is dissipated in the semiconductor switching devices. In recent years, new power semiconductors based on wide band-gap (WBG) devices have been increasingly developed and employed in terms of promising merits including the lower on-state resistance, lower turn-on/off energy, higher capable switching frequency, higher temperature tolerance than conventional Si devices. However, WBG devices also brought new challenges including lower fault tolerance, higher system cost, gate driver challenges, and high dv/dt and resulting increased bearing current in electric machines. This work first proposed a hybrid Si IGBTs + SiC MOSFETs five-level transistor clamped H-bridge (TCHB) inverter which required significantly fewer number of semiconductor switches and fewer isolated DC sources than the conventional cascaded H-bridge inverter. As a result, system cost was largely reduced considering the high price of WBG devices in the present market. The semiconductor switches operated at carrier frequency were configured as Silicon Carbide (SiC) devices to improve the inverter efficiency, while the switches operated at fundamental output frequency (i.e., grid frequency) were constituted by Silicon (Si) IGBT devices. Different modulation strategies and control methods were developed and compared. In other words, this proposed SiC+Si hybrid TCHB inverter provided a solution to ride through a load short-circuit fault. Another special power electronic, multiport converter, was designed for EV charging station integrated with PV power generation and battery energy storage system. The control scheme for different charging modes was carefully developed to improve stabilization including power gap balancing, peak shaving, and valley filling, and voltage sag compensation. As a result, the influence on the power grid was reduced due to the matching between daily charging demand and adequate daytime PV generation. For special machine drives, such as slotless and coreless machines with low inductance, low core losses, typical drive implementations using conventional silicon-based devices are performance limited and also produce large current and torque ripples. In this research, WBG devices were employed to increase inverter switching frequency, reduce current ripple, reduce filter size, and as a result reduce drive system cost. Two inverter drive configurations were proposed and implemented with WBG devices in order to mitigate such issues for 2-phase very low inductance machines. Two inverter topologies, i.e., a dual H-bridge inverter with maximum redundancy and survivability and a 3-leg inverter for reduced cost, were considered. Simulation and experimental results validated the drive configurations in this dissertation. An integrated AC/AC converter was developed for 2-phase motor drives. Additionally, the proposed integrated AC/AC converter was systematically compared with commonly used topologies including AC/DC/AC converter and matrix converters, in terms of the output voltage/current capability, total harmonics distortion (THD), and system cost. Furthermore, closed-loop speed controllers were developed for the three topologies, and the maximum operating range and output phase currents were investigated. The proposed integrated AC/AC converter with a single-phase input and a 2-phase output reduced the switch count to six and resulting in minimized system cost and size for low power applications. In contrast, AC/DC/AC pulse width modulation (PWM) converters contained twelve active power semiconductor switches and a common DC link. Furthermore, a modulation scheme and filters for the proposed converter were developed and modeled in detail. For the significantly increased bearing current caused by the transition from Si devices to WBG devices, advanced modeling and analysis approach was proposed by using coupled field-circuit electromagnetic finite element analysis (FEA) to model bearing voltage and current in electric machines, which took into account the influence of distributed winding conductors and frequency-dependent winding RL parameters. Possible bearing current issues in axial-flux machines, and possibilities of computation time reduction, were also discussed. Two experimental validation approaches were proposed: the time-domain analysis approach to accurately capture the time transient, the stationary testing approach to measure bearing capacitance without complex control development or loading condition limitations. In addition, two types of motors were employed for experimental validation: an inside-out N-type PMSM was used for rotating testing and stationary testing, and an N-type BLDC was used for stationary testing. Possible solutions for the increased CMV and bearing currents caused by the implementation of WGB devices were discussed and developed in simulation validation, including multi-carrier SPWM modulation and H-8 converter topology

The Lazer-McKenna conjecture for an anisotropic planar exponential nonlinearity with a singular source

Given a bounded smooth domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic elliptic problem $$\begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-s\phi_1-4\pi\alpha\delta_q-h(x)\big]\,\,\,\, \,\textrm{in}\,\,\,\,\,\Omega,\\[2mm] \upsilon=0 \qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\quad \textrm{on}\,\ \,\partial\Omega, \end{cases}$$ where $a(x)$ is a positive smooth function, $s>0$ is a large parameter, $h\in C^{0,\gamma}(\overline{\Omega})$, $q\in\Omega$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $\delta_q$ denotes the Dirac measure with pole at point $q$ and $\phi_1$ is a positive first eigenfunction of the problem $-\nabla\big(a(x)\nabla \phi\big)=\lambda a(x)\phi$ under Dirichlet boundary condition in $\Omega$. We show that if $q$ is both a local maximum point of $\phi_1$ and an isolated local maximum point of $a(x)\phi_1$, this problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $q$ and the quantity $\int_{\Omega}a(x)e^{\upsilon_s}\rightarrow8\pi(m+1+\alpha)a(q)\phi_1(q)$ as $s\rightarrow+\infty$, which give a positive answer to the Lazer-McKenna conjecture for this case.Comment: arXiv admin note: text overlap with arXiv:1908.0553

Airport capacity choice under airport-airline vertical arrangements

This study investigates the effects of airport-airline vertical arrangements on airport capacity choices under demand uncertainty. A multi-stage game is analyzed, in which competing airlines contribute to capacity investments and at the same time share airport revenues. Our analytical results suggest that for a profit-maximizing airport, such a vertical arrangement leads to higher capacity although its profit may not be higher. For a welfare-maximizing airport, such an arrangement has no effect on capacity or welfare. Capital cost savings brought by airport-airline cooperation, if any, always leads to higher capacity, higher profit for a profit-maximizing airport, and higher welfare in the case of a welfare-maximizing airport. Numerical simulations reveal that win-win outcomes may be achieved for an airport and its airlines without government intervention