Article Contents
Article Contents

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

• *Corresponding author: Jaakko Kultima
• 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.

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

 Citation:

• 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 $\beta - \beta_B^b$

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 $\beta - \beta_B^b$

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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