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arXiv · 2311.12672

On Neumann-Poincaré operators and self-adjoint transmission problems

Badreddine Benhellal·Konstantin Pankrashkin
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Abstract

We discuss the self-adjointness in $L^2$-setting of the operators acting as $-\nabla\cdot h\nabla$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $Σ$, without explicit assumptions on the sign of $h$. We establish a number of sufficient conditions for the self-adjointness of the operator with $H^s$-regularity for suitable $s\in[1,\frac{3}{2}]$, in terms of the jump value and the regularity and geometric properties of $Σ$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $Σ$, which is new for the Lipschitz setting.

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Badreddine Benhellal, Konstantin Pankrashkin. 2024-04-17. On Neumann-Poincaré operators and self-adjoint transmission problems. https://arxiv.org/abs/2311.12672

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