In this article we investigate spectral properties of the coupling H+V¿, where H=-ia·¿+mß is the free Dirac operator in R3, m>0 and V¿ is an electrostatic shell potential (which depends on a parameter ¿¿R) located on the boundary of a smooth domain in R3. Our main result is an isoperimetric-type inequality for the admissible range of ¿’s for which the coupling H+V¿ generates pure point spectrum in (-m,m). That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman–Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible ¿’s, and we use this to relate the endpoints of the admissible range of ¿’s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.
Over the years, cable-stayed bridges had became more and more popular in mediumand long-span bridge design. Their progress has been paralel to the technological achivements in the design of stays and anchorage devices. At present, the uncertainty in the fatigue resistance of these elements when different stress ranges are applied and the lack of knowledge on the real solicitations of traffic during the service-life are still important. Hence, the fatigue design criteria normally used in stays and anchorages seem to be highly overconservative, leading to unnecessary budget expenses. A computer pro-gram for the simulation of traffic flow over bridges is used in the present work to show a methodology of analysis to derive the most efficient cross-section area of backstays. Based on that. design criteria are summarized to be in the safe side, but still be optimal from the point of view of fatigue action in highway cable-stayed bridges. The results presented are still preliminar due to the simplifications assumed because of the existing lack of data on S-N curves of cables and anchorages. Therefore. in the future up-dating and enhancement of the results will be possible as more experimental data will be available.