Maximal regularity estimates for higher order differential equations with fluctuating coefficients

Abstract

We give the well-posedness conditions in L₂ (-∞, +∞) for the following differential equation:

-y''' + p(x)y' + q(x)y = f(x),
where p and q are continuously differentiable and continuous functions, respectively, and f ∈ L₂(R). Moreover, we prove for the solution y of this equation the following maximal regularity estimate:

‖y'''‖₂ + ‖py'‖₂ + ‖qy‖₂ ≤ C‖f‖₂
(here ‖⋅‖₂ is the norm in L₂(-∞, +∞)). We assume that the intermediate coefficient p is fast oscillating and not controlled by the coefficient q. The sufficient conditions obtained by us are close to necessary ones. We give similar results for the fourth-order differential equation with singular intermediate coefficients.