Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partial\Omega$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partial\Omega$ uniquely minimize the weighted perimeter in $\Omega$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partial\Omega$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $\Omega=\mathbb{R}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $\omega$ is concave and possibly non-smooth.Comment: final version, to appear in Analysis and Geometry in Metric Space

The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations

Particle methods are a ubiquitous tool for solving the Vlasov-Poisson equation in comoving coordinates, which is used to model the gravitational evolution of dark matter in an expanding universe. However, these methods are known to produce poor results on idealized test problems, particularly at late times, after the particle trajectories have crossed. To investigate this, we have performed a series of one- and two-dimensional "Zel'dovich Pancake" calculations using the popular Particle-in-Cell (PIC) method. We find that PIC can indeed converge on these problems provided the following modifications are made. The first modification is to regularize the singular initial distribution function by introducing a small but finite artificial velocity dispersion. This process is analogous to artificial viscosity in compressible gas dynamics, and, as with artificial viscosity, the amount of regularization can be tailored so that its effect outside of a well-defined region - in this case, the high-density caustics - is small. The second modification is the introduction of a particle remapping procedure that periodically re-expresses the dark matter distribution function using a new set of particles. We describe a remapping algorithm that is third-order accurate and adaptive in phase space. This procedure prevents the accumulation of numerical errors in integrating the particle trajectories from growing large enough to significantly degrade the solution. Once both of these changes are made, PIC converges at second order on the Zel'dovich Pancake problem, even at late times, after many caustics have formed. Furthermore, the resulting scheme does not suffer from the unphysical, small-scale "clumping" phenomenon known to occur on the Pancake problem when the perturbation wave vector is not aligned with one of the Cartesian coordinate axes.Comment: 29 pages, 29 figures. Accepted for publication in ApJ. The revised version includes a discussion of energy conservation in the remapping procedure, as well as some interpretive differences in the Conclusions made in response to the referee report. Results themselves are unchange

Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field

Divergent Perturbation Series

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov's method, according to which they are determined by the saddle-point configurations (instantons) of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotic forms are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical schemes for summation of perturbation series are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for 'non-Borel-summable' series. High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD