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Density results of biharmonic functions on symmetric tensor fields and their applications to inverse problems

Divyansh Agrawal, Sombuddha Bhattacharyya, Pranav Kumar

TL;DR

This work proves that three or more biharmonic functions vanishing on an inaccessible boundary portion Γ suffice to densely span smooth symmetric tensor fields up to order 3 in a bounded domain Ω. Building on Runge-type density and a detailed tensor calculus, the authors show that a semilinear biharmonic operator with third- and lower-order nonlinear perturbations is uniquely determined by partial boundary data, despite Γ being arbitrary. The key innovation is a two-tier density proof: a local boundary-vanishing version and a global continuation, which together enable a full partial-data inverse result for nonlinear perturbations of the biharmonic operator in any dimension n ≥ 2. The methods hinge on holomorphic parameter dependence, higher-order linearizations, and careful tensor decompositions to extract coefficient information from boundary measurements, with potential extensions to polyharmonic operators.

Abstract

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric tensor fields up to order three, in a bounded domain. Furthermore, as an application of the density results, in dimension two or higher, we solve a partial data inverse problem for a biharmonic operator with nonlinear anisotropic third and lower order perturbations. For the inverse problem, we take the Dirichlet data to be supported in an arbitrarily small open set of the boundary and measure the Neumann data on the same set. Note that the analogous problem for linear perturbations are still unknown. So far, partial data problems recovering nonlinear perturbations were studied only up to vector fields. The full data analogues of the inverse problem has recently been studied for three or higher dimensions.

Density results of biharmonic functions on symmetric tensor fields and their applications to inverse problems

TL;DR

This work proves that three or more biharmonic functions vanishing on an inaccessible boundary portion Γ suffice to densely span smooth symmetric tensor fields up to order 3 in a bounded domain Ω. Building on Runge-type density and a detailed tensor calculus, the authors show that a semilinear biharmonic operator with third- and lower-order nonlinear perturbations is uniquely determined by partial boundary data, despite Γ being arbitrary. The key innovation is a two-tier density proof: a local boundary-vanishing version and a global continuation, which together enable a full partial-data inverse result for nonlinear perturbations of the biharmonic operator in any dimension n ≥ 2. The methods hinge on holomorphic parameter dependence, higher-order linearizations, and careful tensor decompositions to extract coefficient information from boundary measurements, with potential extensions to polyharmonic operators.

Abstract

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric tensor fields up to order three, in a bounded domain. Furthermore, as an application of the density results, in dimension two or higher, we solve a partial data inverse problem for a biharmonic operator with nonlinear anisotropic third and lower order perturbations. For the inverse problem, we take the Dirichlet data to be supported in an arbitrarily small open set of the boundary and measure the Neumann data on the same set. Note that the analogous problem for linear perturbations are still unknown. So far, partial data problems recovering nonlinear perturbations were studied only up to vector fields. The full data analogues of the inverse problem has recently been studied for three or higher dimensions.
Paper Structure (13 sections, 15 theorems, 100 equations)

This paper contains 13 sections, 15 theorems, 100 equations.

Table of Contents

  1. Introduction and the main results
  2. An application to Inverse Problems
  3. Assumption 1
  4. Preliminaries
  5. Well-posedness of the PDE (\ref{['Dirichlet_Problem']})
  6. Calculus of symmetric tensor fields
  7. Some existing density results using biharmonic functions
  8. Proofs of the density theorems
  9. Reduction of Theorem \ref{['MR1']} to Theorem \ref{['MR2']}.
  10. A local version of Theorem \ref{['MR2']}
  11. The coefficients vanish everywhere
  12. Application to inverse problems
  13. Multi-linearization

Key Result

Theorem 1.1

Let $\mathcal{A}_m, \mathcal{E}_\Gamma$ be as above and $m \geq 2$. The integral identity implies that $\mathcal{A}_{m} \equiv 0$, i.e., the coefficients $A^{(l)}$ vanish identically in $\Omega$ for $l=0,\cdots,3$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • proof : Sketch of proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 20 more