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
\(\nu(\tau)\) 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.
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Dedicated to my friend and colleague Stefan Samko on the occasion of his 70th birthday
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Rabinovich, V. (2013). Potential Type Operators on Weighted Variable Exponent Lebesgue Spaces. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_18
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DOI: https://doi.org/10.1007/978-3-0348-0516-2_18
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