Abstract
We study the heat flow from an open, bounded set D in \(\mathbb {R}^{2}\) with a polygonal boundary ∂D. The initial condition is the indicator function of D. A Dirichlet 0 boundary condition has been imposed on some but not all of the edges of ∂D. We calculate the heat content of D in \(\mathbb {R}^{2}\) at t up to an exponentially small remainder as t ↓ 0.
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Acknowledgments
MvdB was supported by a Leverhulme Trust Emeritus Fellowship EM-2018-011-9. MvdB acknowledges hospitality by the Max Planck Institute for Mathematics, Bonn, and the Mathematical Institute, University of Neuchâtel. KG acknowledges support from the Max Planck Institute for Mathematics, Bonn, from October 2017 to July 2018. The authors wish to thank the referee for helpful suggestions.
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van den Berg, M., Gilkey, P.B. & Gittins, K. Heat Flow from Polygons. Potential Anal 53, 1043–1062 (2020). https://doi.org/10.1007/s11118-019-09797-5
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DOI: https://doi.org/10.1007/s11118-019-09797-5