[Paper Review] A "milder" version of Calder\'on's inverse problem for anisotropic conductivities and partial data

El Maati Ouhabaz|arXiv (Cornell University)|Jan 29, 2015

Numerical methods in inverse problems22 references3 citations

TL;DR

This paper presents a spectral-theoretic proof of unitary equivalence between anisotropic elliptic operators with partial Dirichlet-to-Neumann data, showing that if their partial D-t-N maps coincide for a set with accumulation point, then the operators are unitarily equivalent under Robin, mixed, or Dirichlet boundary conditions. The key result extends prior work by avoiding reliance on unique continuation and operator extension theory, instead using eigenvalue monotonicity and holomorphicity of the D-t-N map.

ABSTRACT

Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data, i.e., data supported in a part of the boundary. We prove positivity, $L^p$-estimates and domination properties for the semigroup associated with this D-t-N operator. Given $L\_a $ and $L\_b$ of the previous type with bounded measurable coefficients $a = \{a\_{kj}, \ a\_k, a\_0 \}$ and $b = \{b\_{kj}, \ b\_k, b\_0 \}$, we prove that if their partial D-t-N operators (with $a\_0$ and $b\_0$ replaced by $a\_0 -\la$ and $b\_0 -\la$) coincide for all $\la$, then the operators $L\_a$ and $L\_b$, endowed with Dirichlet, mixed or Robin boundary conditions are unitary equivalent. In the case of the Dirichlet boundary conditions, this result was proved recently by Behrndt and Rohleder \cite{BR12} for Lipschitz continuous coefficients. We provide a different proof which works for bounded measurable coefficients and other boundary conditions.

Motivation & Objective

• To establish unitary equivalence of anisotropic elliptic operators under partial boundary measurements using spectral theory.
• To provide an alternative proof to Behrndt and Rohleder's result on D-t-N map uniqueness, valid for Robin, mixed, and Dirichlet boundary conditions.
• To weaken regularity assumptions on coefficients, particularly for dimension d = 2, where Lipschitz continuity is not required.
• To show that eigenvalues of the D-t-N operator and the Robin-regularized elliptic operator are strictly monotone in the Robin parameter.

Proposed method

• Define the Dirichlet-to-Neumann (D-t-N) operator with partial data on a subset Γ1 ⊂ ∂Ω, using weak solutions to the elliptic equation La(λ)u = 0.
• Use the holomorphicity of the D-t-N map ⟨NΓ1,a(λ)ϕ, ψ⟩ on C ∖ σ(LD_a) to extend equality of D-t-N maps to a full complex domain.
• Establish strict monotonicity of eigenvalues λμ_a,k with respect to the Robin parameter μ, using the min-max principle and unique continuation.
• Prove that equality of D-t-N maps implies identical spectra and multiplicities for the Robin-regularized operators Lμ_a and Lμ_b.
• Use the limit μ → −∞ to recover the Dirichlet case, showing that eigenvalues of LD_a and LD_b coincide with same multiplicities.
• Construct a unitary map between eigenbases of Lμ_a and Lμ_b, proving unitary equivalence for all boundary conditions.

Experimental results

Research questions

• RQ1 Under what conditions does equality of partial Dirichlet-to-Neumann maps imply unitary equivalence of the underlying elliptic operators?
• RQ2 Can the uniqueness result for Calderón’s inverse problem with partial data be proven without assuming Lipschitz continuity of coefficients?
• RQ3 How does the spectral behavior of the Robin-regularized operator Lμ_a depend on the parameter μ?
• RQ4 What is the role of the D-t-N map’s holomorphicity in extending equality from real to complex λ?
• RQ5 Can the unitary equivalence of operators with different boundary conditions (Robin, mixed, Dirichlet) be established from the same D-t-N data?

Key findings

• If the partial D-t-N maps NΓ1,a(λ) and NΓ1,b(λ) coincide for all λ in a set with accumulation point in ρ(LD_a) ∩ ρ(LD_b), then La and Lb are unitarily equivalent under Robin, mixed, or Dirichlet boundary conditions.
• The eigenvalues of the Robin-regularized operators Lμ_a and Lμ_b are strictly decreasing functions of μ, ensuring injectivity of the spectral correspondence.
• The spectra and multiplicities of Lμ_a and Lμ_b are identical for all μ ∈ ℝ, established via equality of D-t-N maps and eigenvalue monotonicity.
• The limit μ → −∞ recovers the Dirichlet case, showing that λD_a,k = λD_b,k with same multiplicities, implying unitary equivalence of LD_a and LD_b.
• The trace of the kernel of (λ − Lμ_a) equals that of (λ − Lμ_b) for λ ∉ σ(LD_a) = σ(LD_b), confirming spectral matching on the boundary.
• The proof avoids the use of operator extension theory and unique continuation in the main argument, relying instead on spectral theory and holomorphic functional calculus.