C² Regularity of the Surface Tension for the ∇ϕ Interface Model

Abstract

We consider the ∇ϕ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least C2,β for some β > 0. We also prove a fluctuation‐dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite‐volume surface tension to that of its infinite‐volume limit.

Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models

“…The second result of this article establishes the C 2 -regularity of the surface tension σ. It provides a second proof of the result of [12] and removes the Hölder regularity assumption on V ′′ , thus fully resolving the conjecture of [47] and [44, Problem 5.1]. In fact, we show that σ ∈ C 2 (R d ) even if V is uniformly convex and C 1,1 (R)-that is, the surface tension may have more regularity than the interaction potential.…”

“…In this direction, we establish two theorems: In Theorem 1.1, we obtain a quantitative version of the hydrodynamic limit of Funaki and Spohn [46], quantified both over the rate of convergence and the stochastic integrability. In Theorem 1.3, we prove the C 2 -regularity of the surface tension σ under the assumption that the potential is C 1,1 (R), generalizing the result [12] where the regularity is proved for potentials V whose second derivative is Hölder continuous, and solving the conjecture of [47] and [44, Problem 5.1] in full generality.…”

“…The identity (1.5) is known as the fluctuation-dissipation relation, and was recently established by Armstrong and Wu [12] under the assumption that the second derivative of the potential V is Hölder continuous.…”

“…In this article, we are interested in studying this problem in the light of the recent progress in quantitative stochastic homogenization for nonlinear equations [7,6,41,27]. The interplay between homogenization and gradient interface models goes back to the work of [70], and quantitative homogenization methods was the main strategy used in [12]. In these works, the equation being homogenized is the linear, infinite-dimensional Helffer-Sjöstrand equation.…”

“…The fluctuation-dissipation relation in this terminology is known as the commutativity of homogenization and linearization, because it essentially says that the homogenized coefficient for the linearized equation is equal to the coefficient for the linearization of the homogenized equation (see [7,6,41]). The identity (1.5) was established in [12] under the assumption that V ′′ is Hölder continuous, based on a different approach which relied on a quantification of the homogenization of the infinite-dimensional Helffer-Sjöstrand PDE.…”