Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator

Jaakko Kultima; Valery Serov

doi:10.3934/ipi.2022011

Article Contents

2022, Volume 16, Issue 5: 1047-1061. Doi: 10.3934/ipi.2022011

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Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator

Jaakko Kultima^, and
Valery Serov

Research Unit of Mathematical Sciences, P.O. BOX 3000, FIN-90014 University of Oulu, Finland

^*Corresponding author: Jaakko Kultima
^*Corresponding author: Jaakko Kultima

Received: September 2021

Revised: January 2022

Early access: March 2022

Published: October 2022

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Abstract

The inverse backscattering Born approximation for two-dimensional quasi-linear biharmonic operator is studied. We prove the precise formulae for the first nonlinear term of the Born sequence. We prove also that all other terms in this sequence are $ H^t- $functions for any $ t<1 $. These formulae and estimates allow us to conclude that all main singularities of a certain combination of unknown coefficients, in particular, $ L^p- $singularities for $ 2\le p<\infty $, can be uniquely reconstructed using the inverse backscattering Born approximation. In addition, it is shown that the jumps ($ L^{\infty}- $singularities) over smooth curves are uniquely determined by the backscattering data and can be recovered from the Born approximation. We present a numerical method for the reconstruction of these singularities.

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Mathematics Subject Classification: Primary: 35P25, 65M32; Secondary: 35R30, 35J91.

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Figure 1. Combination $ \beta $ consisting three disjoint parts is depicted on the left hand side. In the middle we have the TSVD reconstruction of the function $ \beta_B^b $ and on the right the difference $ \beta - \beta_B^b $

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Figure 2. The second example includes different types of non-linearities compared with the first example. In particular, here the function $ V $ is independent on $ |u| $ and therefore this example highlights that the linear potentials are included in our considerations. In the middle we have the TSVD reconstruction of function $ \beta_B^b $ and on the right the difference $ \beta - \beta_B^b $

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Figure 3. In the third example, only the first order potential $ \overrightarrow{W} $ is presented. The unknown combination $ \beta = -\frac{1}{2}\nabla\cdot \overrightarrow{W} $ (on the left-hand side) in this case has jump-discontinuities over smooth boundaries and as Corollary 1 suggests, such jump-discontinuities of $ \beta $ can be recovered, by using the backscattering Born approximation. In the middle we have the TSVD-reconstruction of function $ \beta^b_B $ and on the right-hand side the difference $ \beta - \beta_B^b $

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