The biharmonic obstacle problem with varying obstacles and a related maximal operator

Abstract

It is well known that maximal functions of various types control various modes of convergence. From the classical Hardy-Littlewood maximal function to certain spherical maximal functions in Harmonic Analysis to maximal operators in Functional Analysis, this is an idea that has been played and replayed throughout analysis. Here, we wish to look at a version of the biharmonic obstacle problem with varying obstacles to again illustrate how a maximal function can control convergence. Our basic idea is to perturb a given fixed obstacle ψ — to the obstacles ψ k K = 1, 2, 3, … — and find conditions that insure that the sequence of solutions {u k } for the pertubed obstacles converge, in an appropriate sense, to the solution u corresponding to the original obstacle. An important feature here is, of course, the types of convergence given and required for the obstacle sequence as well as for the solution sequence. It can happen that if the obstacle sequence convergence is too weak, then the solution sequence may fail to converge to the solution of the expected limit obstacle problem, but rather converges to the solution of a “relaxed” problem. In fact, this seems to be the typical scenario for Γ-convergence methods used to treat sequences of variational problems in general; see [AP] and [DP]. But this is where the idea of using a maximal function, to control things, appears. The basic idea in this context was first proposed in [AV], where a similar problem was considered, but for the polyharmonic case and with the most general data (obstacles) possible. The simplified version presented here is used to highlight this approach with less technical difficulties and for clearity and ease of understanding. We also wish to make some further observations and conjectures.