Surface energy and boundary layers for a chain of atoms at low temperature

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $\beta^{-1}$ goes to zero. Our main results are: (1) As $\beta \to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $\mu_\beta$ and $\nu_\beta$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathrm{surf}$. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $\overline{\mathcal{E}}_\mathrm{surf}$. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in $\beta$

Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature β- 1 goes to zero. Our main results are: (1) As β→ ∞ at fixed positive pressure p> 0 , the Gibbs measures μβ and νβ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals E¯ bulk and E¯ surf. The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of E¯ surf; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in β. © 2020, The Author(s)

완전 비선형 포물 편미분 방정식의 정칙 이론과 그 응용

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 8. 이기암.이 학위 논문에서는 비발산 구조를 갖는 완전 비선형 포물 방정식의 해의 정칙 이론과 그 응용에 대하여 연구하였다. 첫번째 장은 완전 비선형 고른 포물형 및 퇴화된 포물형 방정식의 해의 점근적 행동 양상에 대한 연구이다. 먼저, 포물 방정식의 정규화 된 해가 시간이 흐름에 따라 방정식과 관련된 완전 비선형 타원 작용소의 제 1 고유 함수로 수렴함을 증명하였다. 또한 볼록한 영역에서 오목한 완전 비선형 제차 작용소가 주어졌을때, 포물형 해의 초기 기하적 구조-특정한 오목성(log-concavity, power concavity)-가 보존되는 것을 보였다. 위의 수렴성을 이용하면 제 1 고유 함수 또한 같은 기하적 구조를 가짐을 알 수 있다. 두번째 장에서는 완전 리만 다양체 위에서 완전 비선형 포물 방정식의 해를 다루었는데, 특히 정칙 이론의 초석이 되는 포물형 Harnack 부등식을 증명하였다. 선형 작용소에 대해서는 거리 함수로 정의된 특정한 조건을 가정하고 정칙인 해의 대역적 Harnack 부등식을 얻었다. 또 단면 곡률의 하한을 가지는 다양체에 대해 비선형 작용소의 국소적 Harnack 부등식을 보였다. 마지막으로 Jensen의 sup- and inf-convolution을 이용하여, 연속 해인 viscosity 해에 대한 Harnack 부등식을 증명하였다.Abstract 1 Introduction 1 1.1 Long-time asymptotics for parabolic equations 2 1.2 Parabolic Harnack inequality on Riemannian manifolds 4 2 Preliminaries 8 2.1 Viscosity solutions 8 2.1.1 Uniformly elliptic operator 8 2.1.2 Viscosity solutions 10 2.1.3 Regularity for uniformly elliptic and parabolic equations 11 2.2 Riemannian geometry 12 2.2.1 Variation formulas and Volume comparison 15 2.2.2 Semi-concavity 18 2.2.3 Viscosity solutions on Riemannian manifolds 19 3 Asymptotic behavior of parabolic equations 22 3.1 Uniformly parabolic equations 22 3.1.1 Elliptic eigenvalue problem 22 3.1.2 Long-time asymptotics for uniformly parabolic equations 23 3.1.3 Log-concavity 29 3.2 Degenerate parabolic equations 39 3.2.1 Sub-linear elliptic eigenvalue problems 39 3.2.2 Long-time asymptotics for degenerate parabolic equations 42 3.2.3 Square-root concavity of thepressure 47 4 Harnack inequality on Riemannian manifolds 69 4.1 Harnack inequality for linear parabolic operators 69 4.1.1 ABP-Krylov-Tso type estimate 70 4.1.2 Barrier functions 78 4.1.3 Parabolic version of the Calderon-Zygmund decomposition 90 4.1.4 Proof of parabolic Harnack inequality 94 4.1.5 Weak Harnack inequality 107 4.2 Harnack inequality for nonlinear parabolic operators 110 4.3 Harnack inequality for viscosity solutions 121 4.3.1 Sup-and inf-convolution 121 4.3.2 Proof of parabolic Harnack inequality 132 Abstract (in Korean)Docto