Uniqueness results for inverse Robin problems with bounded coefficient

Abstract

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, $r\textgreater{}n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context

On Some Extremal Problems for Analytic Functions with Constraints on Real or Imaginary Parts

“…Observe that in the present two-dimensional case, Theorems 4, 5 still hold whenever K is a more general set of finite positive Lebesgue measure [4,11,23]. This should also be the case for Theorem 6 if S is of finite positive Lebesgue measure, see [7].…”

Magnetic moment estimation and bounded extremal problems

“…This last reference assumes the dimension is at least 3, but the planar case (which is our concern here) may be treated in the same way. The result we need is also explicitly stated and formally proved in [4,Prop. 5.2], which deals exclusively with dimension 2 (but more general equations).…”

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

“…Results on uniqueness and stability for recovering the inclusion and/or impedance has been studied in recent manuscripts. 15,16 In Bacchelli, 15 it is proven that roughly speaking two Cauchy pairs are enough to uniquely determine the boundary of the inclusion provided the currents are linearly independent and nonnegative. To prove the uniqueness the author uses techniques for classical solution to Laplace's equation which requires that be a C 1, function and Γ 0 is class C 2, for some 0 < < 1.…”