Abstract
We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set \({\Omega }\subset \mathbb {R}^{m}\) of finite Lebesgue measure \(0<|{\Omega }|<\infty \) with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.
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Acknowledgements
MvdB and TK acknowledge support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9, and the Swiss National Science Foundation respectively. DB was supported by the LabEx PERSYVAL-Lab GeoSpec (ANR-11-LABX-0025-01) and ANR SHAPO (ANR-18-CE40-0013).
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Berg, M.v.d., Bucur, D. & Kappeler, T. On Efficiency and Localisation for the Torsion Function. Potential Anal 57, 571–600 (2022). https://doi.org/10.1007/s11118-021-09928-x
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DOI: https://doi.org/10.1007/s11118-021-09928-x
Keywords
- Torsion function
- First Dirichlet eigenfunction
- Schrödinger operator
- Dirichlet boundary condition
- Localisation
- Efficiency