Assumptions and Notations :

• $\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$,

• $K'_0$ is the normal derivative of the single layer potential associated to the Laplace equation:

$K'_0 u(x)=\int_{\Gamma}u(y) \partial_{n(x)} g_0(x-y) ds(y)$

avec $g_0(x-y)=\frac{1}{2\pi} \ln(\frac{1}{|x-y|})$.

The question:

If we choose the real number $\lambda$ such that the operator $\lambda Id - K'_0$ is Fredholm on the Sbolev space $H^{-\frac 1 2}(\Gamma)$, can we then find an explicit expression of a regularizer for that operator, which will be denoted by $(\lambda Id - K?_0)'$? (at least, for some particular domains as those with cornes (polyhedral domain), using the local Mellin transformation I suppose).

Here, $A'$ a regularizer of $A$ means a bilateral regularizer, i.e. such that

$A' A = Id + K_l$, $A A' = Id + K_r$, $K_l$ et $K_r$ are compacts operators and $Id$ is the identity map).

Note that if $\Gamma$ is smooth, then $K'_0$ defines a compact operator on $H^{-\frac 1 2}(\Gamma)$ and the answer is trivial.