The low-temperature thermal conductivity in polycrystalline graphene is theoretically studied. The contributions from three branches of acoustic phonons are calculated by taking into account scattering on sample borders, point defects and grain bound
aries. Phonon scattering due to sample borders and grain boundaries is shown to result in a $T^{alpha}$-behaviour in the thermal conductivity where $alpha$ varies between 1 and 2. This behaviour is found to be more pronounced for nanosized grain boundaries. PACS: 65.80.Ck, 81.05.ue, 73.43.Cd
Given the standard Gaussian measure $gamma$ on the countable product of lines $mathbb{R}^{infty}$ and a probability measure $g cdot gamma$ absolutely continuous with respect to $gamma$, we consider the optimal transportation $T(x) = x + abla varphi(
x)$ of $g cdot gamma$ to $gamma$. Assume that the function $| abla g|^2/g$ is $gamma$-integrable. We prove that the function $varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {det}_2(I + D^2 varphi) exp bigl(mathcal{L} varphi - 1/2 | abla varphi|^2 bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $varphi$.
Let $A subset mathbb{R}^d$, $dge 2$, be a compact convex set and let $mu = varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ u = varrho_1 dx$ be a probability measure on $B_r := {xcolon |x| le r}$ e
quivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ u = mu circ T^{-1}$ and $T = phi cdot {rm n}$, where $phicolon A to [0,r]$ is a continuous potential with convex sub-level sets and ${rm n}$ is the Gauss map of the corresponding level sets of $phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $phi$ the level sets of $phi$ are driven by the Gauss curvature flow $dot{x}(s) = -s^{d-1} frac{varrho_1(s {rm n})}{varrho_0(x)} K(x) cdot {rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.
