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

Electronic Journal of Differential Equations

article

Jesús Ildefonso Díaz¹  Tatiana A. Shaposnikova²  Maria N. Zubova² 

[1] Instituto de Matematica Interdisciplinar Universidad Complutense Madrid;Faculty of Mechanics and Mathematics Moscow State University Moscow

关键词: Critically scaled homogenization;    asymmetric particles;    dynamic boundary conditions;    Holder continuous reactions;    strange term;    nonlocal monotone operator.;   

DOI  :  10.58997/ejde.2022.52

学科分类：数学（综合）

来源: Texas State University

【 摘 要 】

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 Holder 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.

【 授权许可】

CC BY