Abstract
Let D be a non-empty open subset of \(\mathbb {R}^{m}, m\ge 2\), with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε ↓ 0 of the \(L^{\infty }\) norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that D ∖ K is a non-trap domain.
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MvdB acknowledges support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9.
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Berg, M.v.d., Carroll, T. On the Torsion Function with Mixed Boundary Conditions. Potential Anal 55, 277–284 (2021). https://doi.org/10.1007/s11118-020-09857-1
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DOI: https://doi.org/10.1007/s11118-020-09857-1