Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Maximum principle for higher order operators in general domains

We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.