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An obstacle problem for the p−elastic energy

Marius Mueller

Dienstag, 16. November 2021 14:15Uhr

In this talk we seek to minimize the p-elastic curvature energy E(u) := \int_graph(u) |κ|^p ds among all graphs u ∈ W^2,p (0, 1)∩ W_0^1,p(0, 1) that satisfy the obstacle constraint u(x) ≥ ψ(x) for all x ∈ [0, 1]. Here ψ ∈ C^0([0, 1]) is an obstacle function. The energy functional imposes three major challenges that we need to overcome:

  1. Lack of coercivity.

  2. Loss of regularity on the coincidence set {u = ψ}.

  3. (For p > 2:) Degeneracy of the Euler-Lagrange equation.

We will develop methods to examine all three phenomena. A key ingredient for this analysis goes back to L. Euler: One can find a substitution that makes the Euler-Lagrange equation elliptic.

Finally, we are able to show sharp existence (and non-existence) results and discuss the optimal regularity of minimizers.