On the Torsion Function with Mixed Boundary Conditions

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 DK is a non-trap domain.

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Acknowledgments

MvdB acknowledges support by the Leverhulme Trust through Emeritus Fellowship EM-2018-011-9.

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Correspondence to M. van den Berg.

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Berg, M.v.d., Carroll, T. On the Torsion Function with Mixed Boundary Conditions. Potential Anal (2020). https://doi.org/10.1007/s11118-020-09857-1

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Keywords

  • Torsion function
  • Dirichlet boundary condition
  • Neumann boundary condition

Mathematics Subject Classification (2010)

  • 35J25
  • 35J05
  • 35P15