Recent Results on a Generalization of the Laplacian^†

In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a function W(x ₁, ..., x_d ) = Σ^d_k=1 W_k (x_k ), where each W_k : ℝ → ℝ is a right continuous with left limits and strictly increasing function. Using W, we construct the generalized laplacian ℒW = Σ^d_i=1 ∂xi ∂wi, where ∂wi is a generalized differential operator induced by the function W_i . We present results on spectral properties of ℒW, Sobolev spaces induced by ℒW(W-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations and stochastic homogenization.

W-Sobolev space; generalized Laplacian; homogenization; partial differential equations