We consider the inverse boundary value problem of the simultaneous determination of the coefficients <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> and q of the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-\text{ div }(\sigma \nabla u)+qu = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mspace width="0.333333em" /> <mtext>div</mtext> <mspace width="0.333333em" /> <mo stretchy="false">(</mo> <mi>σ</mi> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>q</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> of the boundary <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> of a domain <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We assume that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> and q are a-priori known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with curved interfaces. We prove that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> and q can be uniquely determined in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> from the knowledge of the local map.

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

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2026年4月17日

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Niall Donlon,
Romina Gaburro

摘要

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $\sigma $ $σ$ and q of the equation $-\text{ div }(\sigma \nabla u)+qu = 0$ $- div (σ \nabla u) + q u = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given locally on a non-empty curved portion $\Sigma $ $Σ$ of the boundary $\partial \Omega $ $\partial Ω$ of a domain $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ , with $n\ge 3$ $n \geq 3$ . We assume that $\sigma $ $σ$ and q are a-priori known to be a piecewise constant matrix-valued and scalar function, respectively, on a given partition of $\Omega $ $Ω$ with curved interfaces. We prove that $\sigma $ $σ$ and q can be uniquely determined in $\Omega $ $Ω$ from the knowledge of the local map.