An obstacle problem for the p-elastic energy

TL;DR

This paper studies an obstacle problem for generalized p-elastic energy in graphical curves, analyzing solution regularity, existence, and uniqueness, especially focusing on degeneracy and obstacle effects.

Contribution

It provides new insights into the regularity and existence of solutions for the obstacle problem involving p-elastic energy, including sharp results for the p-elastica case.

Findings

Solutions may have flat parts where curvature vanishes.

Conditions are identified for existence and nonexistence of solutions.

Uniqueness results are established for symmetric minimizers in the p-elastica case.

Abstract

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler--Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.