A "milder" version of Calderón's inverse problem for anisotropic conductivities and partial data

2015/01/29 by El Maati Ouhabaz, Ouhabaz, El Maati
Mathematics · #Numerical methods in inverse problems #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1501.07364

Abstract

Given a general symmetric elliptic operator L_a := ∑_k,,j=1^d \p_k (a_kj \p_j) + ∑_k=1^d a_k \p_k - \p_k(a_k .) + a_0we 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 \citeBR12 for Lipschitz continuous coefficients. We provide a different proof which works for bounded measurable coefficients and other boundary conditions.

Related

• Strict positivity for the principal eigenfunction of elliptic operators with various boundary conditions
• Partial data inverse problems for quasilinear conductivity equations
• The Calderón problem is an inverse source problem
• Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators
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