Stability estimates for parabolic problems with Wentzell boundary conditions


Of concern is the uniformly parabolic problemut = div (A ∇ u), u (0, x) = f (x), ut + β ∂νA u + γ u - q β ΔLB u = 0, for x ∈ Ω ⊂ RN and t ≥ 0. Here A = {ai j (x)}i j is a real, hermitian, uniformly positive definite N × N matrix; β, γ ∈ C (over(Ω, -)) with β > 0; q ∈ [0, ∞) and ∂νA u is the conormal derivative of u with respect to A: and everything is sufficiently regular. The solution of this well-posed problem depends continuously on the ingredients of the problem, namely, A, β, γ, q, f. This is shown using semigroup methods in [G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum, in press]. More precisely, if we have a sequence of such problems with solutions un, and if An → A, βn → β, etc. in a suitable sense, then un → u, the solution of the limiting problem. The abstract analysis associated with operator semigroup theory gives this conclusion, but no rate of convergence. Determining how fast the convergence of the solutions is requires detailed estimates. Such estimates are provided in this paper. © 2008 Elsevier Inc. All rights reserved.

Publication Title

Journal of Differential Equations