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I want to know if there is a way to solve mixed Neumann-Dirichlet boundary value problem using Greens function. For instance, let us assume an infinite boundary with reflecting surface ($\partial \Omega_r$) but a small portion of the boundary is absorbing ($\partial \Omega_a$). I want to derive the PDF of particle motion in such an environment using a diffusion differential equation. I searched for literature and get Narrow escape problem but there they assume that the absorbing boundary is quite small as compared to the reflective one. What if the absorbing boundary is not that small, is there a reference that deals specifically with mixed boundary value problems.

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Actually, if you look at the derivation of the narrow escape PDE, for instance in a paper written by Xinfu Chen & Carey Caginalp. The "size" of the window $\varepsilon > 0 $ can be as large as possible. The name "narrow escape" refers to the fact that $\varepsilon$ is tiny, whence various perturbation technique can be used.

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  • $\begingroup$ Is there some literature that deals with mixed boundary conditions, in contrast to boundary being totally absorbing or reflecting. Or is it that mixed boundary conditions are always referred to as narrow escape problems in literature? $\endgroup$
    – Userhanu
    Commented Jul 22, 2020 at 7:15
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    $\begingroup$ From my understanding, the narrow escape problem specificly refers to the search of the solution of the boundary value problem $-\Delta u = -2$ in $\Omega$, with zero Dirichlet boundary condition on a small piece of the $\partial \Omega$, and zero Neumann boundary condition on the complementary large piece of $\partial \Omega$. $\endgroup$
    – Fei Cao
    Commented Jul 22, 2020 at 17:53
  • $\begingroup$ @Userhanu I recently wrote and submitted a very short paper (micro-article) about the narrow escape problem (which is not on the arxiv), I think it would be good for an introduction to this type of problems, if you are interested in take a look I can send the pdf file via email $\endgroup$
    – Fei Cao
    Commented Jan 8, 2021 at 23:46

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