Continuous dependence in hyperbolic problems with Wentzell boundary conditions


Let Ω be a smooth bounded domain in ℝN and let Lu = ∑ j,κ=1N∂xj (a jκ(x)∂xκu) ; in Ω and Lu + β(x) ∑j, κ=1Najκ(x) ∂xj u nκ +γ(x)u - qβ(x) ∑ j, κ=1N-1 ∂τκ (b jκ(x)∂τj u)= 0; on ∂Ωdefine a generalized Laplacian on Ω with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, -S2, on a suitable Hilbert space. If we have a sequence of such operators S0; S1; S2; ... with corresponding coefficients φn =(a jκ(n), b jκ(n), βn, γn, qn) satisfying φn → φ0 uniformly as n → ∞, then un(t) → u0(t) where un satisfies i dun/dt = S nm un, or d2un/dt 2+ Sn2m un = 0, or d 2un/dt2 + F(Sn) dun/dt + Sn2m un = 0, for m = 1, 2, initial conditions independent of n, and for certain nonnegative functions F. This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.

Publication Title

Communications on Pure and Applied Analysis