Conditions for maximal regularity of solutions to fourth-order differential equations

Abstract

This article investigates a fourth-order differential equation defined in a Hilbert space, with an unbounded intermediate coefficient and potential. The key distinction from previous research lies in the fact that the intermediate term of the equation does not obey to the differential operator formed by its extreme terms. The study establishes that the generalized solution to the equation is maximally regular, if the intermediate coefficient satisfies an additional condition of slow oscillation. A corresponding coercive estimate is obtained, with the constant explicitly expressed in terms of the coefficients’ conditions. Fourth-order differential equations appear in various models describing transverse vibrations of homogeneous beams or plates, viscous flows, bending waves, and etc. Boundary value problems for such equations have been addressed in numerous works, and the results obtained have been extended to cases with smooth variable coefficients. The smoothness conditions imposed on the coefficients in this study are necessary for the existence of the adjoint operator. One notable feature of the results is that the constraints only apply to the coefficients themselves; no conditions are placed on their derivatives. Secondly, the coefficient of the lowest order in the equation may be zero, moreover, it may not be unbounded from below.