Recent Results on a Generalization of the Laplacian - Simas, Alexandre B; Valentim, Fábio J (Trends in Computational and Applied Mathematics, 2015)
In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a functionW(x_1,…,x_d) = _k=1^d 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 L_W = _i=1^d _x_i_W_i, where _W_i is a generalized differentialoperator induced by the function W_i.We present results on spectral properties of L_W, Sobolev spaces induced by L_W (W-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations andstochastic homogenization.