The p-elastic energy of a curve f: I to R^2,
is given by
E_p(f)= \int_I |\kappa|^p ds ,
with $\kappa$ the curvature of f and s the arc-length parameter. Here p takes values strictly between 1 and infinity and for p=2 one recovers the Bernoulli model of an elastic rod.
In this talk we consider the obstacle problem obtained minimizing the p-elastic energy on graphs constrained to stay above a given obstacle. We discuss existence, uniqueness and symmetry of minimizers. The main question we address in the talk is: Which is the main cause of the loss of regularity of minimizers, the presence of obstacle or the degeneracy of the Euler–Lagrange equation?
This is joint work with Marius Müller (Freiburg University), Shinya Okabe (Tohoku University) and
Kensuke Yoshizawa (Tohoku University).