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Boundary determination of coefficients appearing in a perturbed weighted $p$-Laplace equation

Nitesh Kumar, Tanmay Sarkar, Manmohan Vashisth

Abstract

We study an inverse boundary value problem associated with $p$-Laplacian which is further perturbed by a linear second order term, defined on a bounded set $Ω$ in $\R^n, n\geq 2$. We recover the coefficients at the boundary from the boundary measurements which are given by the Dirichlet to Neumann map. Our approach relies on the appropriate asymptotic expansion of the solution and it allows one to recover the coefficients pointwise. Furthermore, by considering the localized Dirichlet-to-Neumann map around a boundary point, we provide a procedure to reconstruct the normal derivative of the coefficients at that boundary point.

Boundary determination of coefficients appearing in a perturbed weighted $p$-Laplace equation

Abstract

We study an inverse boundary value problem associated with -Laplacian which is further perturbed by a linear second order term, defined on a bounded set in . We recover the coefficients at the boundary from the boundary measurements which are given by the Dirichlet to Neumann map. Our approach relies on the appropriate asymptotic expansion of the solution and it allows one to recover the coefficients pointwise. Furthermore, by considering the localized Dirichlet-to-Neumann map around a boundary point, we provide a procedure to reconstruct the normal derivative of the coefficients at that boundary point.
Paper Structure (13 sections, 5 theorems, 96 equations)

This paper contains 13 sections, 5 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. $\varepsilon$-expansion of the solution for perturbed $p$-Laplacian problem
  3. Case-1: $p>2$
  4. Case-2: $1<p<2$.
  5. $\varepsilon$-expansion of DN-map
  6. Proof for the Theorem \ref{['thm_1']}
  7. Reconstruction of $\sigma$
  8. Reconstruction of $\gamma$
  9. Case: $p>2$
  10. Case: $1<p<2$
  11. Proof for the Theorem \ref{['thm_2']}
  12. Reconstruction of normal derivative of $\sigma$
  13. Reconstruction of normal derivative of $\gamma$

Key Result

Theorem 1.1

(Reconstruction of coefficients) Suppose $p>1$ with $p\neq 2$ and $\Omega \subseteq\mathbb{R}^n,n\geq 2$ is a bounded domain with smooth boundary $\partial\Omega$. Furthermore, assume that the coefficients are continuous on $\overline{\Omega}$ and bounded below by a positive constant. Then the coeff

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Weak solutions
  • Proposition 2.2: Strong solution
  • Proposition 4.1
  • proof