On the Neumann Laplacian in nonuniformly collapsing strips

Abstract

Consider the Neumann Laplacian in the region below the graph of εg(x), for a positive smooth function g : [a, ∞) → R with both g (x)/g(x) and (g (x)/g(x)) bounded. As ε → 0 such region collapses to [a, ∞) and an effective operator is found, which has Robin boundary conditions at a. Then we recover (under suitable assumptions in the case of unbounded g) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for g diverging less than exponentially.