Strict convexity of the free energy for non-convex gradient models at moderate $β$

Strict Convexity of the Surface Tension for Non-convex Potentials

We study gradient models on the lattice $\mathbb{Z}^d$ with non-convex interactions. These Gibbs fields (lattice models with continuous spin) emerge in various branches of physics and mathematics. In quantum field theory they appear as massless field theories. Even though our motivation stems from considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models, our attention here is mostly devoted to interfaces, with the gradient field as an \emph{effective} interface interaction. In this case we prove the strict convexit…

Scaling limit and strict convexity of free energy for gradient models with non-convex potential

We consider gradient models on the lattice $\mathbb{Z}^d$. These models serve as effective models for interfaces and are also known as continuous Ising models. The height of the interface is modelled by a random field with an energy which is a non-convex perturbation of the quadratic interaction. We are interested in the Gibbs measure with tilted boundary condition $u$ at inverse temperature $β$ of this model. In [AKM16], [Hil16] and [ABKM19] the authors show that for small tilt $u$ and large inverse temperature $β$ the surface tension is strictly convex, where the limit is taken on a subseque…

Displacement convexity and minimal fronts at phase boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition, are strictly convex in the sense of displacement convexity under a natural change of variables. We use this to show that in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals, and, in the multi-component setting, jointly displacement co…

非量子化奇异Liouville型方程爆破解的非退化性与带奇异源的Onsager涡旋模型平均场熵的凸性

Non degeneracy of blow-up solutions of non-quantized singular Liouville-type equations and the convexity of the mean field entropy of the Onsager vortex model with singular sources

本文证明了奇异平均场方程在正则或非量子化奇异源爆破解点处泡泡解的非退化性，推广了已有结果，并由此得到带奇异源的Onsager涡旋模型中一类二维区域熵在大能量极限下的严格凸性。

Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction

In a recent paper we studied an equation (called the "simple equation") introduced by one of us in 1963 for an approximate correlation function associated to the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $ρ$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $ρ(e)$ for the density as a function of the energy $e$. We had conjectured that $ρ(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(ρ)$. We had also conjec…