Error estimates and lifespan of effective boundary conditions for 2-dimensional optimally aligned coatings

Abstract This paper provides error estimates between solutions of the “full model” of the heat equation on a coated body and solutions of the “effective model” obtained by Chen et al. (2012) [9] . We also estimate the length of time period (called lifespan) during which the effective model is a good approximation of the full model.

[1]  Alain Prignet,et al.  Nonlinear reinforcement problems with right-hand side in L1 , 2008, Asymptot. Anal..

[2]  S. Nazarov,et al.  Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues , 2006 .

[3]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[4]  Xuefeng Wang,et al.  Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body , 2008 .

[5]  Kaijun Zhang,et al.  REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER , 2011 .

[7]  A. Friedman.  Reinforcement of the principal eigenvalue of an elliptic operator , 1980 .

[8]  Cody Pond,et al.  Effective Boundary Conditions on Coated Bodies , 2011 .

[9]  Keddour Lemrabet,et al.  The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation , 1996, SIAM J. Appl. Math..

[10]  Xuefeng Wang,et al.  Using effective boundary conditions to model fast diffusion on a road in a large field , 2017 .

[11]  Xuefeng Wang,et al.  Asymptotic behavior of Robin problem for heat equation on a coated body , 2012 .

[12]  M. Vivaldi,et al.  Reinforcement problems for variational inequalities on fractal sets , 2015 .

[13]  Jingyu Li,et al.  Bulk-surface coupling: Derivation of two models , 2021, Journal of Differential Equations.

[14]  Jingyu Li.  Asymptotic behavior of solutions to elliptic equations in a coated body , 2009 .

[15]  Alexis Auvray,et al.  Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers , 2018 .

[16]  Analysis of curvature influence on effective boundary conditions , 2002 .

[17]  Huicong Wang.  Effective Boundary Conditions for the Heat Equation with Interior Inclusion , 2020 .

[18]  Xuefeng Wang,et al.  The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body , 2014 .

[19]  Reinforcement problems in elasto-plasticity , 1980 .

[20]  S. Nazarov,et al.  Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems , 2006 .

[21]  normalsizesf Wang XueFeng.  Effective boundary conditions of diffusion equations on domains containing thin layers , 2016 .

[22]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[23]  Avner Friedman,et al.  Reinforcement problems for elliptic equations and variational inequalities , 1980 .

[24]  Giuseppe Buttazzo,et al.  Reinforcement problems in the calculus of variations (*) (*)Financially supported by a national research project of the Italian Ministry of Education. , 1986 .

[25]  Robert V. Kohn,et al.  Reinforcement by a thin layer with oscillating thickness , 1987 .

[26]  Xuefeng Wang,et al.  Suppression of the Dirichlet Eigenvalues of a Coated Body , 2006, SIAM J. Appl. Math..

[27]  Huicong Li.  Effective boundary conditions of the heat equation on a body coated by functionally graded material , 2015 .

[28]  Serguei A. Nazarov,et al.  Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions , 2011, Networks Heterog. Media.

[29]  Linear and semi-linear reinforcement problems by thin layers , 2003 .

[30]  Xuefeng Wang,et al.  Effective Boundary Conditions Resulting from Anisotropic and Optimally Aligned Coatings: The Two Dimensional Case , 2012 .

[31]  Sergei A. Nazarov,et al.  Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands , 2021 .

[32]  L. Rosenhead.  Conduction of Heat in Solids , 1947, Nature.