Regularity for fully nonlinear integro-differential operators with regularly varying kernels

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{sigma, beta}$ satisfying $$ K_{sigma,beta}(y)asymp frac{ 2-sigma}{|y|^{n+sigma}}left( logfrac{2}{|y|^2}right)^{beta(2-sigma)}quad mbox{near zero} $$ with respect to $sigmain(0,2)$ close to $2$ (for a given $betainmathbb R$), where the regularity estimates do not blow up as the order $ sigmain(0,2)$ tends to $2.$

Impact of the low solar abundance on the ages of globular clusters

We present the result of our investigation on the impact of the low Solar abundance of Asplund and collaborators (2004) on the derived ages for the oldest star clusters based on isochrone fittings. We have constructed new stellar models and corresponding isochrones using this new solar mixture with a proper Solar calibration. We have found that the use of the Asplund et al. (2004) metallicity causes the typical ages for old globular clusters in the Milky Way to be increased roughly by 10%. Although this may appear small, it has a significant impact on the interpretation for the formation epoch of Milky Way globular clusters. The Asplund et al. (2004) abundance may not necessarily threaten the current concordance cosmology but would suggest that Milky Way globular clusters formed before the reionization and before the main galaxy body starts to build up. This is in contrast to the current understanding on the galaxy formation.