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Extension of Boundary Control method to elliptic and parabolic problems, and its application to the Calderon problem

Dimitra Kyriakopoulou

TL;DR

This work extends the Boundary Control (BC) method, originally developed for hyperbolic inverse problems, to elliptic and parabolic inverse boundary value problems using Schulze’s edge calculus, with the aim of unifying inverse boundary value problems under BC. It develops the necessary algebraic and microlocal framework—graded operator algebras, principal and edge symbols, localization, Calkin algebra, and noncommutative analysis—to establish BC-based uniqueness for elliptic problems and Volterra-type parabolic problems, all within an edge-/cone-degenerate setting. The Calderón problem is treated directly within this BC framework by reformulating it as an edge problem, identifying Fredholm regimes (notably γ ≠ $ rac{1}{2}$, $ rac{3}{2}$), and showing that identical Dirichlet-to-Neumann data force an isometric diffeomorphism between manifolds via a Gelfand-transform–driven coordinatization. Overall, the paper offers a pathway for unifying hyperbolic, elliptic, and parabolic inverse boundary value problems through BC methodology, leveraging noncommutative and operator-algebraic perspectives to access direct elliptic results without hyperbolization. $$ abla \cdot (\sigma \nabla u)=0,\quad \Lambda_\sigma[f]=\sigma \frac{\partial u}{\partial \nu}\Big|_{\partial\mathcal{M}}$$ and intricate edge-symbol analyses underpin these advances, with implications for geometric reconstruction from boundary data.

Abstract

We show that Boundary Control method, a method for hyperbolic inverse problems, is also capable of dealing directly with certain classes of elliptic and parabolic Inverse Boundary Value Problems; thus pointing towards Boundary Control method potentially constituting a means of unification of Inverse Boundary Value Problems. As an application we show that the Calderon problem can be dealt with directly via Boundary Control method, i.e. without reduction of the elliptic problem to a 'hyperbolized' problem.

Extension of Boundary Control method to elliptic and parabolic problems, and its application to the Calderon problem

TL;DR

This work extends the Boundary Control (BC) method, originally developed for hyperbolic inverse problems, to elliptic and parabolic inverse boundary value problems using Schulze’s edge calculus, with the aim of unifying inverse boundary value problems under BC. It develops the necessary algebraic and microlocal framework—graded operator algebras, principal and edge symbols, localization, Calkin algebra, and noncommutative analysis—to establish BC-based uniqueness for elliptic problems and Volterra-type parabolic problems, all within an edge-/cone-degenerate setting. The Calderón problem is treated directly within this BC framework by reformulating it as an edge problem, identifying Fredholm regimes (notably γ ≠ , ), and showing that identical Dirichlet-to-Neumann data force an isometric diffeomorphism between manifolds via a Gelfand-transform–driven coordinatization. Overall, the paper offers a pathway for unifying hyperbolic, elliptic, and parabolic inverse boundary value problems through BC methodology, leveraging noncommutative and operator-algebraic perspectives to access direct elliptic results without hyperbolization. and intricate edge-symbol analyses underpin these advances, with implications for geometric reconstruction from boundary data.

Abstract

We show that Boundary Control method, a method for hyperbolic inverse problems, is also capable of dealing directly with certain classes of elliptic and parabolic Inverse Boundary Value Problems; thus pointing towards Boundary Control method potentially constituting a means of unification of Inverse Boundary Value Problems. As an application we show that the Calderon problem can be dealt with directly via Boundary Control method, i.e. without reduction of the elliptic problem to a 'hyperbolized' problem.

Paper Structure

This paper contains 11 sections, 19 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Proofs on BC method extensions
  3. Proof on BC for elliptic problems
  4. Facts to be used in the proof
  5. Lemmas
  6. Proof of Theorem \ref{['ell thm']}
  7. Proof on BC for parabolic problems
  8. Facts to be used in the proof
  9. Proof on Theorem \ref{['par thm']}
  10. The Calderon Problem
  11. Proof of Theorem \ref{['Calderon']}

Key Result

Theorem 1.1

The BC method can be used to directly prove uniqueness arguments for inverse elliptic BVPs, when the corresponding edge problem is elliptic in the sense of Schulze’s pseudo-differential operator edge calculus (i.e. both the principal interior symbol and the edge symbol are invertible).

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.1
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 10 more