## Layer potential techniques for the narrow escape problem.(English)Zbl 1252.35116

Summary: The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with Dirichlet boundary condition on a small absorbing part of the boundary and Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems, we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.

### MSC:

 35C20 Asymptotic expansions of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 92B05 General biology and biomathematics 35P15 Estimates of eigenvalues in context of PDEs 35J61 Semilinear elliptic equations 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
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### References:

 [1] Ammari, H.; Garnier, J.; Kang, H.; Lee, H.; Sølna, K., The Mean escape time for a narrow escape problem with multiple switching gates, Multiscale model. simul., 9, 817-833, (2011) · Zbl 1231.82048 [2] Ammari, H.; Kang, H., Polarization and moment tensors with applications to inverse problems and effective medium theory, Applied mathematical sciences, vol. 162, (2007), Springer-Verlag New York · Zbl 1220.35001 [3] Ammari, H.; Kang, H.; Lee, H., Layer potential techniques in spectral analysis, Mathematical surveys and monographs, vol. 153, (2009), Amer. Math. Soc. Providence, RI · Zbl 1167.47001 [4] Berezhkovskii, A.M.; Barzykin, A.V., Extended narrow escape problem: boundary homogenization-based analysis, Phys. rev. E, 82, 011114, (2010) [5] Cheviakov, A.; Ward, M.; Straube, R., An asymptotic analysis of the Mean first passage time for narrow escape problems: part II: the sphere, Multiscale model. simul., 8, 836-870, (2010) · Zbl 1204.35030 [6] Costabel, M.; Penzel, F.; Schneider, R., Error analysis of a boundary element collocation method for a screen problem in $$\mathbb{R}^3$$, Math. comp., 58, 575-586, (1992) · Zbl 0765.65108 [7] Gadylʼshin, R.R., Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions, Differ. equ., 35, 540-551, (1999) · Zbl 1158.35399 [8] Holcman, D., Diffusion in cellular microdomains: application to synapses, (), 1-26 · Zbl 1177.92006 [9] Holcman, D.; Schuss, Z., Escape through a small opening: receptor trafficking in a synaptic membrane, J. stat. phys., 117, 975-1014, (2004) · Zbl 1087.82018 [10] Holcman, D.; Schuss, Z., Diffusion escape through a cluster of small absorbing windows, J. phys. A, 41, 155001, (2008) · Zbl 1138.35379 [11] Holcman, D.; Schuss, Z., Diffusion through a cluster of small windows and flux regulation in microdomains, Phys. lett. A, 372, 3768-3772, (2008) · Zbl 1220.92008 [12] Kolokolnikov, T.; Titcombe, M.; Ward, M.J., Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps, European J. appl. math., 16, 161-200, (2005) · Zbl 1090.35070 [13] Nédélec, J.C., Acoustic and electromagnetic equations. integral representations for harmonic problems, Applied mathematical sciences, vol. 144, (2001), Springer-Verlag New York · Zbl 0981.35002 [14] Pillay, S.; Ward, M.J.; Peirce, A.; Kolokolnikov, T., An asymptotic analysis of the Mean first passage time for the narrow escape problems: part I: two-dimensional domain, Multiscale model. simul., 8, 803-835, (2009) · Zbl 1203.35023 [15] Saranen, J.; Vainikko, G., Periodic integral and pseudodifferential equations with numerical approximation, Springer monographs in mathematics, (2001), Springer-Verlag Berlin [16] Schuss, Z.; Singer, A.; Holcman, D., The narrow escape problem for diffusion in cellular microdomains, Proc. natl. acad. sci. USA, 104, 16098-16103, (2007) [17] Singer, A.; Schuss, Z., Activation through a narrow opening, SIAM J. appl. math., 68, 98-108, (2007) · Zbl 1147.60040 [18] Singer, A.; Schuss, Z.; Holcman, D.; Eisenberg, R.S., Narrow escape, I, J. stat. phys., 122, 437-463, (2006) · Zbl 1149.82335 [19] Singer, A.; Schuss, Z.; Holcman, D., Narrow escape, part II: the circular disk, J. stat. phys., 122, 465-489, (2006) · Zbl 1149.82333 [20] Singer, A.; Schuss, Z.; Holcman, D., Narrow escape. III. non-smooth domains and Riemann surfaces, J. stat. phys., 122, 491-509, (2006) · Zbl 1149.82334 [21] Singer, A.; Schuss, Z.; Holcman, D., Narrow escape and leakage of Brownian particles, Phys. rev. E, 78, 051111, (2008), 8 pp [22] Stephan, E.P., Boundary integral equations for screen problems in $$\mathbb{R}^3$$, Integral equations operator theory, 10, 236-257, (1987) · Zbl 0653.35016 [23] Ward, M.J.; Keller, J.B., Strong localized perturbations of eigenvalue problems, SIAM J. appl. math., 53, 770-798, (1993) · Zbl 0778.35081 [24] Ward, M.J.; Henshaw, W.D.; Keller, J.B., Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAM J. appl. math., 53, 799-828, (1993) · Zbl 0778.35082 [25] Ward, M.J.; Van De Velde, E., The onset of thermal runaway in partially insulated or cooled reactors, IMA J. appl. math., 48, 53-85, (1992) · Zbl 0796.35060
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