A strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape
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Date
2022-07-18
Authors
Diaz, Jesus Ildefonso
Shaposnikova, Tatiana
Zubova, Maria
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Hölder nonlinear σ(u). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on L2(0,T), independently of the regularity of σ.
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Keywords
Critically scaled homogenization, Asymmetric particles, Dynamic boundary conditions, Hölder continuous reactions, Strange term, Nonlocal monotone operator
Citation
Díaz, J. I., Shaposhnikova, T. A., & Zubova, M. N. (2022). A strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape. <i>Electronic Journal of Differential Equations, 2022</i>(52), pp. 1-32.
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Attribution 4.0 International
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This work is licensed under a Creative Commons Attribution 4.0 International License.