Rank-(n – 1) convexity and quasiconvexity for divergence free fields

The CAST experiment at CERN (European Organization of Nuclear Research) searches for axions from the sun. The axion is a pseudoscalar particle that was motivated by theory thirty years ago, with the intention to solve the strong CP problem. Together with the neutralino, the axion is one of the most promising dark matter candidates. The CAST experiment has been taking data during the last two years, setting an upper limit on the coupling of axions to photons more restrictive than from any other solar axion search in the mass range below 0.1 eV. In 2005 CAST will enter a new experimental phase extending the sensitivity of the experiment to higher axion masses. The CAST experiment strongly profits from technology developed for high energy physics and for X-ray astronomy: A superconducting prototype LHC magnet is used to convert potential axions to detectable X-rays in the 1-10 keV range via the inverse Primakoff effect. The most sensitive detector system of CAST is a spin-off from space technology, a Wolter I type X-ray optics in combination with a prototype pn-CCD developed for ESA's XMM-Newton mission. As in other rare event searches, background suppression and a thorough shielding concept is essential to improve the sensitivity of the experiment to the best possible. In this context CAST offers the opportunity to study the background of pn-CCDs and its long term behavior in a terrestrial environment with possible implications for future space applications. We will present a systematic study of the detector background of the pn-CCD of CAST based on the data acquired since 2002 including preliminary results of our background simulations.Comment: 11 pages, 8 figures, to appear in Proc. SPIE 5898, UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XI

Two optimization problems in thermal insulation

We consider two optimization problems in thermal insulation: in both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The total mass of the insulating material is prescribed.. The first problem deals with the case in which a given heat source is present, while in the second one there are no heat sources and the goal is to have the slowest decay of the temperature. In both cases an optimal distribution of the insulator around the thermal body exists; when the body has a circular symmetry, in the first case a constant heat source gives a constant thickness as the optimal solution, while surprisingly this is not the case in the second problem, where the circular symmetry of the optimal insulating layer depends on the total quantity of insulator at our disposal. A symmetry breaking occurs when this total quantity is below a certain threshold. Some numerical computations are also provided, together with a list of open questions.Comment: 11 pages, 7 figures, published article on Notices Amer. Math. Soc. is available at http://www.ams.org/publications/journals/notices/201708/rnoti-p830.pd

Sobolev regularity and an enhanced Jensen inequality

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to a minimization problem with constraints, where regularity of minimizers cannot be deduced from the Euler-Lagrange equation.Comment: 10 page