Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).

We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains and establish a unified proof that works for different regimes of hole-cell ratios, which is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and yields correctors and error estimates for vanishing hole-cell ratios. For a positive volume fraction of holes, the approach is just the standard oscillating test function method; for a vanishing volume fraction of holes, we study asymptotic behaviors of properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris, and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin-type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that of an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole space with infinitely many interfaces, a setting that is out of the standard stationary ergodic framework.