We consider a random, uniformly elliptic coefficient field a on the lattice ℤd. The distribution ⟨.⟩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) |2satisfy optimal annealed estimates which are L2 and L1, respectively, in probability, i.e., they obtained bounds on ⟨| ∇ x G (t,x,y)|2 ⟩1⁄2 and ⟨| ∇ x ∇y G(t,x,y)|⟩ .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., Lp in probability for all p<∞. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for ⟨| ∇ x G (x,y)|2 ⟩1⁄2 and ⟨| ∇ x ∇y G(x,y)|⟩.
In this review article we present an overview on some a priori estimates in L p , p > 1, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two L p -bounds, p > 2, for the solution of the associated Dirichlet problem, obtained in correspondence with two different sign assumptions. Then, putting together the above mentioned bounds and using a duality argument, we extend the estimate also to the case 1 < p < 2, for each sign assumption, and for a data in L p ∩ L 2 .