Asymptotic behaviour of a class of degenerate elliptic-parabolic operators: a unitary approach

Abstract

Abstract.We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equa-The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators (r h ≡ 0), G-convergence for parabolic operators (r h ≡ 1), singular perturbations of an elliptic operator (a h ≡ a and r h → r, possibly r ≡ 0). Mathematics Subject Classification. 35J15, 35K10, 35M10, 45J45.

$G$-convergence of elliptic and parabolic operators depending on vector fields

“…there exists a suitable a such that ρ h ∂ t + div X (a h (x, t, X)) G-converges to ρ∂ t + div X (a(x, t, X)) independently of the sequence (ρ h ) h (see Section 2.3 for details). This technique was introduced in [28], using a regularity result to get the compactness of the solutions, and then applied in [29], where ρ h may also be negative. As by-product, we get a compactness result for purely parabolic operators and a compactness result for purely elliptic (independent of t) operators.…”