An inverse problem for inhomogeneous Signorini obstacle

Ziyao Zhao

arxiv: 2605.15091 · v1 · pith:VAYHD7Q6new · submitted 2026-05-14 · 🧮 math.AP

An inverse problem for inhomogeneous Signorini obstacle

Ziyao Zhao This is my paper

Pith reviewed 2026-05-15 03:18 UTC · model grok-4.3

classification 🧮 math.AP

keywords obstacleproblemsignoriniboundaryinversemeasurementsappliesarbitrary

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The pith

The shape and obstacle function of an inhomogeneous Signorini problem are uniquely determined by boundary measurements on an arbitrary open subset for both scalar and elasticity versions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Signorini problem models situations where a physical system, like a membrane or elastic body, cannot penetrate an obstacle and makes contact according to a specific function. This creates a variational inequality rather than a standard equation. The inverse problem tries to recover the unknown obstacle from external measurements. This work establishes a uniqueness result: knowing the solution on any small open piece of the outer boundary is enough to determine both the geometry of the obstacle and the contact function that describes it. The result covers the basic scalar case and extends to the more involved elasticity setting with vector displacements and stresses.

Core claim

both the shape of the obstacle and the obstacle function can be uniquely determined from solution measurements taken on an arbitrary open subset of the boundary. This result applies to both the scalar and elasticity versions of the Signorini problem.

Load-bearing premise

The Signorini problem is well-posed in the inhomogeneous setting, and boundary measurements on an arbitrary open subset suffice for uniqueness without additional regularity or geometric assumptions on the obstacle.

read the original abstract

This paper investigates the inverse problem of determining a general Signorini obstacle using boundary measurements. We demonstrate that both the shape of the obstacle and the obstacle function can be uniquely determined from solution measurements taken on an arbitrary open subset of the boundary. This result applies to both the scalar and elasticity versions of the Signorini problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Uniqueness for recovering inhomogeneous Signorini obstacle shape and function from arbitrary open boundary data, but the unique continuation step across the free boundary looks like the main risk.

read the letter

The paper proves that both the shape of a general inhomogeneous obstacle and the obstacle function itself are uniquely determined by solution measurements on any nonempty open subset of the boundary. The result covers the scalar Signorini problem and the elasticity version. This is the central claim and the main advance over earlier work that stayed with homogeneous obstacles. If the argument holds, it widens the class of inverse problems that can be treated in contact mechanics settings. The abstract presents the result cleanly as a direct consequence of the boundary data, without extra geometric assumptions on the obstacle. That framing is straightforward and worth checking. The potential soft spot is the passage from local boundary traces to global uniqueness. The solution satisfies the elliptic system only off the contact set, and it coincides with the unknown obstacle on a positive-measure portion of the domain. Unique continuation therefore has to cross the free boundary, which typically has only C^{1,α} regularity. Standard Carleman estimates or analyticity tools do not apply automatically once the inhomogeneity is present and two candidate solutions may differ on the contact set. The paper must control this difference carefully; without seeing the details in Sections 3–4 it is not obvious that the argument closes. No numerical tests or explicit examples appear in the abstract, so the result stays purely theoretical. This work is for people already working on inverse problems for variational inequalities and obstacle problems. A specialist in the subfield would want to see the proof to decide whether the extension is solid. I would send it to peer review rather than desk-reject it, because the claim is substantive and the only way to settle the unique-continuation issue is to examine the full argument.

Referee Report

1 major / 1 minor

Summary. The paper establishes a uniqueness result for the inverse problem of recovering an inhomogeneous Signorini obstacle. It claims that both the shape of the obstacle and the obstacle function itself are uniquely determined from solution measurements on an arbitrary open subset of the boundary. The result is stated for both the scalar Signorini problem and its vectorial counterpart in linear elasticity.

Significance. If the uniqueness argument holds, the result would strengthen the theory of inverse problems for variational inequalities by showing that partial boundary data suffice to recover both geometric and functional aspects of the obstacle without additional assumptions on regularity or geometry. The extension from the homogeneous to the inhomogeneous case and from scalar to elasticity settings is a positive contribution, provided the technical obstacles around the free boundary are fully resolved.

major comments (1)

[Sections 3–4] The central uniqueness claim requires unique continuation for the difference of two solutions across the unknown contact set, where the solution coincides with the obstacle and the free boundary has at most C^{1,α} regularity. Standard Carleman estimates or analyticity arguments for elliptic equations do not apply directly in this setting; the manuscript must specify (in the uniqueness proof, presumably Sections 3–4) how the inhomogeneous term and the variational inequality are controlled to obtain the continuation property from an arbitrary open boundary arc.

minor comments (1)

[Abstract and Theorem 1.1] The abstract states the result for 'an arbitrary open subset' but does not clarify whether the subset must be connected or satisfy any geometric condition relative to the free boundary; this should be made explicit in the statement of the main theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard well-posedness of the Signorini variational inequality and unique continuation properties for the underlying PDE, without introducing new free parameters or entities.

axioms (1)

domain assumption The Signorini problem admits a unique solution in the appropriate Sobolev space for given obstacle data.
Invoked implicitly to set up the forward map from obstacle to boundary measurements.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. ... If Λ₁ = Λ₂, then O₁ = O₂ and ϕ₁ = ϕ₂.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3. ... Then u is a constant function in U.

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