Potential Type Operators on Weighted Variable Exponent Lebesgue Spaces

Abstract

We consider double-layer potential type operators acting in weighted variable exponent Lebesgue space $L^{p(.)}(\Gamma,w)$ on some composed curves with oscillating singularities. We obtain a Fredholm criterion for operators $A=aI+bD_{g.\Gamma}:L^{p(.)}(\Gamma,w)\rightarrow L^{p(.)}(\Gamma,w) \; {\rm where}{D_{g,\Gamma}}$ is the operator of the form

$${D_{g,\Gamma}{u(t)}}=\frac{1}{\pi}\int_{\Gamma}\frac{g(t,\tau)(\nu(\tau),\tau-t)u(\tau)dl_{t}}{|t-\tau|^{2}},t \in \Gamma $$

$\nu(\tau)$ is the inward unit normal vector to Γ at the point $\tau \in \Gamma \setminus \mathcal{F},dl_{\tau}$ is the oriented Lebesgue measure on $\tau ,\mathcal{F}$ is the set of the nodes, $a,b:\Gamma\rightarrow\mathbb{C},g:\Gamma\times\Gamma \rightarrow \mathbb{C}$ are a bounded functions with oscillating discontinuities at the nodes only.

We give applications of such operators to the Dirichlet and Neumann problems with boundary function in $L^{p(.)}(\Gamma,w)$ for domains with boundaries having a finite set of oscillating singularities.

Mathematics Subject Classification (2010). 31A10, 31A25.