A nonlinear Liouville theorem for fractional equations in the Heisenberg group

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as established in \cite{FGMT}. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space \mathbb H^n\times \mathbbR^+

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.Comment: 33 pages. arXiv admin note: text overlap with arXiv:0709.1103 by other author

Nonlocal elliptic hemivariational inequalities

This paper is devoted to the existence of solutions for the hemivariational inequalities involving fractional Laplace operator by means of the well-known surjectivity result for pseudomonotone mappings

The existence of periodic solutions of a two dimensional Lattice

We consider a two dimensional lattice coupled with nearest neighbor interaction potential of power type. The existence of infinite many periodic solutions is shown by using minimax methods

The Brezis-Nirenberg type problem involving the square root of the Laplacian

We establish existence and non-existence results to the Brezis-Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition

Optimal Gear-Shifting of a Wet-Type Two-Speed Dual-Brake Transmission for an Electric Vehicle

In improving the efficiency of powertrain systems and ride comfort for electric vehicles (EVs), the transmission model is required to enable more accessible and more straightforward control of such vehicles. In this study, a wet-type, two-speed, dual-brake transmission system, as well as a new electromechanical clutch actuator, is presented for EVs. A new coordinated optimal shifting control strategy is then introduced to avoid sharp jerks during shifting processes in the transmission system. Based on a state-space model of the electromechanical clutch actuator and dual-brake transmission, we develop a linear quadratic regulator strategy by considering ride comfort and sliding friction work to obtain optimal control trajectories of the traction and shifting motors under model-based control. Simulations and bench tests are carried out to verify the performance of the proposed control laws. Results of the proposed coordinated control strategy show that noticeable improvements in terms of vehicle jerk and friction energy loss are achieved compared with an optimal control scheme only for the shifting motor as the input

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type

Performance of first insert coil with REBCO CICC sub-size cable exceeding 6 kA at 21 T magnetic field

The Institute of Plasma Physics at the Chinese Academy of Sciences is developing the REBCO cable in conduit conductor (CICC) technology for applications in next-generation nuclear fusion devices. The aim is to develop a CICC comprised of six REBCO sub-cables to satisfy the requirements of operation with a current of around 40 kA and a peak field of up to 20 T. To qualify the performance of the sub-size REBCO cable to be used in the CICC, two 25-turn insert solenoids have been designed, manufactured and tested at a current exceeding 6 kA subjected in a background field supplied by a water-cooled resistive magnet. The insert solenoid, wound from a 11.5 m long REBCO CORC$^{®}$ cable, was designed to investigate its current carrying capacity under high field and electromagnetic (EM) load at 4.2 K. Tests were performed under a background magnetic field up to 18.5 T, resulting in a peak magnetic field on the innermost layer turns of around 21.1 T at an operating current of 6.3 kA. The effects of operation with cyclic EM loads were tested by repeated current ramps to around 95% of the critical current. Moreover, the V–I characteristics were measured at 77 K and the self-field, to check the effects from warm-up and cool-down (WUCD) cycles between room temperature and 77 K with liquid nitrogen. The results show no obvious degradation after dozens of high-current test cycles in background fields ranging from 10 T to 18.5 T. The insert solenoid demonstrates the stable operation of the REBCO sub-size cable for CICC with EM loads of about 90 kN m$^{−1}$ and WUCD cycles between room temperature and 77 K. These promising results indicate the potential of this technology for further applicationsin particular, for full-size CICC for high-performance fusion magnets