On the very weak solvability of the beam equation

Abstract

We get some necessary and sufficient conditions for the very weak solvability of the beam equation stated in terms of powers of the distance to the boundary, accordingly to the boundary condition under consideration. We get a L 1 -estimate by using an abstract result due to Crandall and Tartar. Applications to some nonlinear perturbed equations and to the eventual positivity of the solution of the parabolic problems are also given.

A nonlinear bilaplacian equation with hinged boundary conditions and very weak solutions: analysis and numerical solution

“…This idea was originally introduced in [2] by Haïm Brezis in the seventies (see also [4]). More recently, for higher order equations, in [7] some new results proving that the class of L 1 Loc ( ) data for which the existence and uniqueness of a very weak solution can be obtained is, in general, larger than L 1 ( : δ), which is actually the optimal class for second order equations. For instance, for the beam equation with Dirichlet boundary conditions it is proved that the optimal class of data is the space L 1 ( : δ 2 ).…”