Annealed estimates on the Green functions and uncertainty quantification

2014/09/01 by Gloria, Antoine, Marahrens, Daniel
#35B65 #35J08 #35J15 #60H25 #60K37 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.0569

Abstract

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on ℝ^d for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality and for periodic coefficients, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. As a main application we obtain optimal estimates on the fluctuations of solutions of linear elliptic PDEs with "noisy" diffusion coefficients, an uncertainty quantification result. As a direct corollary of the decay estimates we also prove that for these classes of coefficients the Hölder exponent of the celebrated De Giorgi-Nash-Moser theory can be taken arbitrarily close to 1 in the large (that is, away from the singularity).

Related

• Annealed estimates on the Green function
• On annealed elliptic Green function estimates
• Boundary estimates and Green function's expansion for elliptic systems with random coefficients
• Green's function for elliptic systems: moment bounds
• Asymptotic expansion of the annealed Green's function and its derivatives