In this talk we study the biharmonic Alt-Caffarelli problem. This means we minimize a functional in H²(Ω) consisting of two competing summands — a bending term and an adhesive term.
The bending term is the energy associated to the biharmonic operator. The adhesive term penalizes the volume of the positivity set. Furthermore, we fix positive boundary values u₀ > 0. Minimizers need to find a balance: “avoidance of bending” vs. “a large negativity region” — competing interests!
While smoothness of minimizers is ensured apart from their zero level set, regularity is lost once the zero level is touched — a free boundary! In 2D we understand the (optimal) global regularity of solutions and the geometry of their free boundary: Minimizers lie in C² and the free boundary consists of finitely many simple disjoint C²-curves.
A crucial observation is that the gradient of a solution does not vanish on the free boundary. During the talk I hope to convince you of this fact.
This talk reports on findings in arxiv.org/abs/2001.04914. For more about the optimal regularity: arxiv.org/abs/2103.00303.