On Neumann-Poincaré operators and self-adjoint transmission problems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911842789490688 |
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| author | Benhellal, Badreddine Pankrashkin, Konstantin |
| author_facet | Benhellal, Badreddine Pankrashkin, Konstantin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12672 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Neumann-Poincaré operators and self-adjoint transmission problems Benhellal, Badreddine Pankrashkin, Konstantin Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis 81Q10, 81V05, 35P15, 58C40 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. |
| title | On Neumann-Poincaré operators and self-adjoint transmission problems |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs Functional Analysis 81Q10, 81V05, 35P15, 58C40 |
| url | https://arxiv.org/abs/2311.12672 |