A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the uniqueness question for the semilinear elliptic boundary value problem -{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of {\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of {\Omega} being a ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case {\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3

Measurement of the ratio of branching fractions BR(B0 -> K*0 gamma)/BR(Bs0 -> phi gamma) and the direct CP asymmetry in B0 -> K*0 gamma

The ratio of branching fractions of the radiative B decays B0 -> K*0 gamma and Bs0 phi gamma has been measured using an integrated luminosity of 1.0 fb-1 of pp collision data collected by the LHCb experiment at a centre-of-mass energy of sqrt(s)=7 TeV. The value obtained is BR(B0 -> K*0 gamma)/BR(Bs0 -> phi gamma) = 1.23 +/- 0.06(stat.) +/- 0.04(syst.) +/- 0.10(fs/fd), where the first uncertainty is statistical, the second is the experimental systematic uncertainty and the third is associated with the ratio of fragmentation fractions fs/fd. Using the world average value for BR(B0 -> K*0 gamma), the branching fraction BR(Bs0 -> phi gamma) is measured to be (3.5 +/- 0.4) x 10^{-5}. The direct CP asymmetry in B0 -> K*0 gamma decays has also been measured with the same data and found to be A(CP)(B0 -> K*0 gamma) = (0.8 +/- 1.7(stat.) +/- 0.9(syst.))%. Both measurements are the most precise to date and are in agreement with the previous experimental results and theoretical expectations.Comment: 21 pages, 3 figues, 4 table