On annealed elliptic Green's function estimates

Marahrens, Daniel, and Otto, Felix. "On annealed elliptic Green's function estimates." Mathematica Bohemica 140.4 (2015): 489-506. <http://eudml.org/doc/271826>.

@article{Marahrens2015,
abstract = {We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb \{Z\}^d$. The distribution $\langle \cdot \rangle $ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\smash\{\langle |\nabla _x G(t,x,y)|^2\rangle ^\{\{1\}/\{2\}\}\}$ and $\langle |\nabla _x \nabla _y G(t,x,y)|\rangle $. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors extended these elliptic bounds to higher moments, i.e., $L^p$ in probability for all $p<\infty $. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for $\langle |\nabla _x G(x,y)|^2\rangle ^\{\{1\}/\{2\}\}$ and $\langle |\nabla _x \nabla _y G(x,y)|\rangle $.},
author = {Marahrens, Daniel, Otto, Felix},
journal = {Mathematica Bohemica},
keywords = {stochastic homogenization; elliptic equation; Green’s function on $\mathbb \{Z\}^d$; annealed estimate},
language = {eng},
number = {4},
pages = {489-506},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On annealed elliptic Green's function estimates},
url = {http://eudml.org/doc/271826},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Marahrens, Daniel
AU - Otto, Felix
TI - On annealed elliptic Green's function estimates
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 4
SP - 489
EP - 506
AB - We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb {Z}^d$. The distribution $\langle \cdot \rangle $ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\smash{\langle |\nabla _x G(t,x,y)|^2\rangle ^{{1}/{2}}}$ and $\langle |\nabla _x \nabla _y G(t,x,y)|\rangle $. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors extended these elliptic bounds to higher moments, i.e., $L^p$ in probability for all $p<\infty $. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for $\langle |\nabla _x G(x,y)|^2\rangle ^{{1}/{2}}$ and $\langle |\nabla _x \nabla _y G(x,y)|\rangle $.
LA - eng
KW - stochastic homogenization; elliptic equation; Green’s function on $\mathbb {Z}^d$; annealed estimate
UR - http://eudml.org/doc/271826
ER -