A uniqueness result in the inverse problem for the anisotropic Schrödinger type equation from local measurements

Abstract We consider the inverse boundary value problem of the simultaneous determination of the coefficients $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> and q of the equation $$-\text{ div }(\sigma \nabla u)+qu = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mspace/> <mml:mtext>div</mml:mtext> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> of the boundary $$\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> of a domain $$\Omega \subset \mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , with $$n\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . We assume that $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> and q are a-priori known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> with curved interfaces. We prove that $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> and q can be uniquely determined in $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> from the knowledge of the local map.