Willmore obstacle problems under Dirichlet boundary conditions

TL;DR

This work analyzes obstacle problems for Willmore-type energies under Dirichlet boundary conditions, first in the one-dimensional elastica setting and then for surfaces of revolution. By exploiting explicit symmetric elastica solutions, a priori energy bounds, and a dual variational framework, the authors prove existence of minimisers under universal energy thresholds ($4c_0^2$ in 1D and $4\pi$ for Willmore surfaces), establish high-regularity ($W^{3,\infty}$ with $u'''\in BV$) and derive a sharp universal bound on admissible obstacles ($\psi(x)\le 1.1890464540\ldots$) for the symmetric case, plus nonexistence results beyond this bound. They extend the analysis to non-symmetric data with stronger smallness assumptions and formulate obstacle problems for surfaces of revolution, obtaining analogous existence and regularity results, along with detailed constructions of admissible obstacles (including large-$\alpha$ regimes) and connections to hyperbolic elastica. Overall, the paper provides a rigorous framework for Willmore obstacle problems under Dirichlet conditions, clarifies when solvability holds, and delivers concrete obstacles and examples illustrating the sharpness of the energy-based criteria. The results have potential implications for the study of constrained Willmore surfaces and variational models in geometric analysis and materials science.

Abstract

We consider obstacle problems for the Willmore functional in the class of graphs of functions and surfaces of revolution with Dirichlet boundary conditions. We prove the existence of minimisers of the obstacle problems under the assumption that the Willmore energy with the unilateral constraint is below a universal bound. We address the question whether such bounds are necessary in order to ensure the solvability of the obstacle problems. Moreover, we give several instructive examples of obstacles such that minimisers exist.

Figures (9)

• Figure 1: $\hat{u}_{2.3}$ (left) and $\hat{u}_{c_0}$ (right)
• Figure 2: $\hat{u}_{1.0}$ (left) and $\hat{u}_{c_0/2}$ (right)
• Figure 3: Any smooth function built from the brown circular arcs and a part either of the green or of the blue $\cosh$ and slightly enlarged near $\pm 1$ is an admissible obstacle for $\alpha=0.9$.
• Figure 4: Admissible obstacles (up to enlarging near $\pm 1$) for $\alpha=2$ (left) and $\alpha=1$ (right).
• Figure 5: Two admissible obstacles (up to enlarging near $\pm 1$) for $\alpha=0.99$.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['theorem:1.1']}
• Remark 2.3
• Remark 2.4