Regularity of the free boundary in the biharmonic obstacle problem

Abstract

In this article we use a flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible domain, we derive the C 1,α-regularity of the free boundary in a small ball centred at the origin. From the C 1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally C 3,α up to the free boundary, and therefore C 2,1. In the end we study an example, showing that in general C 2, 1 2 is the best regularity that a solution may achieve in dimension n ≥ 2.

A stabilised finite element method for the plate obstacle problem

“…[13,9], but it cannot belong to H 4 (Ω ). The exact solutions given in [6,1] seem to indicate that the smoothness threshold is C 2,1/2 (Ω ) or H 7/2−ε (Ω ), ε > 0. The solution to the clamped plate bending problem is more regular if the obstacle is elastic.…”

The Biharmonic Alt-Caffarelli Problem in 2D

“…The classical first-order formulation can be understood as a variational Dirichlet problem with 'adhesion' term. More exactly, the energy the authors consider is given by E AC puq :" ˆΩ |∇u| 2 dx `|tx P Ω : upxq ą 0u| where u P W 1,2 pΩq is such that u ´u0 P W 1,2 0 pΩq for some given sufficently regular positive function u 0 . Here, | ¨| denotes the Lebesgue measure and Ω Ă R n is some sufficiently regular domain.…”

“…The measure penalization can be understood as an adhesion to the zero level: Indeed, the lattice operations on W 1,2 imply that each minimizer of E AC is nonnegative. Given this, a minimizer u divides Ω into two regions, namely tu " 0u, the so-called nodal set, and tu ą 0u.…”

“…The article was trendsetting for the study of fourth order free boundary problems and gave way to striking recent results in this field, cf. [1], [31], [32].…”

“…Let us remark that for smooth Ω, one can remove "loc" in the W 3,2´β regularity statement, see Section 9 for details where also Navier boundary conditions are discussed. Let us formally motivate the term 1 |∇u| dH 1 in (1.2). It can be seen as a 'derivative' of the |tu ą 0u|-term of the energy in the following way: By [15,Prop.3,Sect.3.3.4] one has that for each f P C 8 pR n q with nonvanishing gradient,…”