Abstract
The mixed boundary value problem for the Poisson (or Pontryagin–Andronov–Vitt) equation in three dimensions is more complicated than that in two dimensions, primarily because the singularity of Neumann’s function for a regular domain is more complicated than (6.4).
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Notes
- 1.
For non-smooth \(p_0\) the integral is not uniformly bounded. For example, for \(p_0 = \delta ({\varvec{x}}-{\varvec{x}}_0)\) we have \(\displaystyle {\int _{\Omega }}N({\varvec{x}},{\varvec{y}})p_0({\varvec{y}})\, d{\varvec{y}}= N({\varvec{x}},{\varvec{x}}_0)\), which becomes singular as \({\varvec{x}}\rightarrow {\varvec{x}}_0\). However, this is an integrable singularity, and as such it does not affect the leading order asymptotics in \(\delta \).
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Holcman, D., Schuss, Z. (2018). Narrow Escape in \(\mathbb R^3\). In: Asymptotics of Elliptic and Parabolic PDEs. Applied Mathematical Sciences, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-319-76895-3_8
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DOI: https://doi.org/10.1007/978-3-319-76895-3_8
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