The Calderón inverse problem for isotropic quasilinear conductivities

An inverse problem for a quasilinear convection--diffusion equation

“…One of these difficulties comes from the fact that the parabolic equation studied here is not self adjoint as opposed to the self adjoint elliptic equation studied in [4]. This makes some of the symmetries present in the latter work to disappear as is apparent already from the statement of our Proposition 1.1 compared to the analogous proposition in [4]. Secondly, the form of the geometric optics solutions here are rather different from the complex geometric optics solutions constructed in [4].…”

“…A second order linearization method was considered by Sun and Uhlmann in [45] while the idea of higher order linearization was fully utilized by Kurylev, Lassas and Uhlmann in [33] to solve challenging inverse problems for hyperbolic equations. Without being exhaustive, we refer the reader for example to the works [13,17,28,33,35,37] that study inverse problems for nonlinear hyperbolic equations, [7,14,19,24,25,31,32,36,44] for some results concerning semilinear elliptic equations as well as [3,4,5,16,22,29,40,43,45] for results on quasilinear elliptic equations. All these works are based on the linearization method.…”

“…In order to prove Proposition 1.1, we consider a specific class of geometric optics solutions whose products concentrate near arbitrary points x 0 ∈ Ω and t 0 ∈ (0, T ). This idea is inspired by the approach of [4] where a similar construction was carried out in the context of determining a nonlinear conductivity in an elliptic equation. Nevertheless, we would like to mention that for our parabolic problem there are several technical differences compared to the elliptic problem studied in [4] that we will sketch as follows.…”

“…This idea is inspired by the approach of [4] where a similar construction was carried out in the context of determining a nonlinear conductivity in an elliptic equation. Nevertheless, we would like to mention that for our parabolic problem there are several technical differences compared to the elliptic problem studied in [4] that we will sketch as follows. One of these difficulties comes from the fact that the parabolic equation studied here is not self adjoint as opposed to the self adjoint elliptic equation studied in [4].…”

“…Nevertheless, we would like to mention that for our parabolic problem there are several technical differences compared to the elliptic problem studied in [4] that we will sketch as follows. One of these difficulties comes from the fact that the parabolic equation studied here is not self adjoint as opposed to the self adjoint elliptic equation studied in [4]. This makes some of the symmetries present in the latter work to disappear as is apparent already from the statement of our Proposition 1.1 compared to the analogous proposition in [4].…”