An inverse Signorini obstacle problem

TL;DR

The paper tackles the inverse Signorini obstacle problem for isotropic elasticity, showing that a single boundary measurement of displacement and normal stress on an open boundary subset can uniquely determine the obstacle up to a natural obstruction. It develops a BV/set-of-finite-perimeter framework and uses unique continuation, Green’s formulas on sets of finite perimeter, and careful boundary analysis to compare two obstacles and derive contradictions unless they coincide. The results cover both the scalar Laplace Signorini problem and the elasticity system, providing a rigorous obstruction-based characterization of when unique solvability holds. This work advances inverse problems for differential inequalities and provides techniques potentially extensible to broader free-boundary problems and differential-inequality settings.

Abstract

We study the inverse problem of determining a Signorini obstacle from boundary measurements for the isotropic elasticity system. We prove that the obstacle can be uniquely determined by a single measurement of displacement and normal stress for the Signorini problem on an open subset of the boundary up to a natural obstruction. In addition to considering the Signorini problem, we develop techniques that can be used to study inverse problems for general differential inequalities.

Figures (2)

• Figure 1: Boundary measurements of the displacement $\boldsymbol{u}$ on $\partial \Omega$ resulting from rotations cannot distinguish the size of a round Signorini obstacle $O$, in which case the stress tensor is identically zero. The right figure illustrates the displacement vector $\boldsymbol{u}$ resulting from a (infinitesimal) rotation with respect to the center of the obstacle. The displacement on $\partial O$ is perpendicular to the normal direction so the Signorini contact conditions on $\partial O$ are valid.
• Figure 2: An illustration of the sets $G_0$ and $\mathcal{V}$.

Theorems & Definitions (38)

• Theorem 1
• remark 1.1
• Theorem 2
• remark 1.2
• lemma 2.1
• proof
• lemma 2.2
• proof
• lemma 2.3
• proof