The Periodic Unfolding Method: Theory and Applications to Partial Differential Problems

Description

This is the first book on the subject of the periodic unfolding method (originally called "éclatement périodique" in French), which was originally developed to clarify and simplify many questions arising in the homogenization of PDE's. It has since led to the solution of some open problems. Written by the three mathematicians who developed the method, the book presents both the theory as well as numerous examples of applications for partial differential problems with rapidly oscillating coefficients: in fixed domains (Part I), in periodically perforated domains (Part II), and in domains with small holes generating a strange term (Part IV). The method applies to the case of multiple microscopic scales (with finitely many distinct scales) which is connected to partial unfolding (also useful for evolution problems). This is discussed in the framework of oscillating boundaries (Part III). A detailed example of its application to linear elasticity is presented in the case of thin elastic plates (Part V). Lastly, a complete determination of correctors for the model problem in Part I is obtained (Part VI). This book can be used as a graduate textbook to introduce the theory of homogenization of partial differential problems, and is also a must for researchers interested in this field. Front Matter ....Pages i-xv Front Matter ....Pages 1-3 Unfolding operators in fixed domains (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 5-59 Advanced topics for unfolding (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 61-97 Homogenization in fixed domains (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 99-149 Front Matter ....Pages 151-153 Unfolding operators in perforated domains (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 155-198 Homogenization in perforated domains (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 199-235 A Stokes problem in a partially porous medium (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 237-253 Front Matter ....Pages 255-257 Partial unfolding: a brief primer (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 259-261 Oscillating boundaries (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 263-279 Front Matter ....Pages 281-283 Unfolding operators: the case of “small holes” (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 285-295 Homogenization in domains with “small holes” (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 297-353 Front Matter ....Pages 355-357 Homogenization of an elastic thin plate (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 359-398 Front Matter ....Pages 399-401 The scale-splitting operators revisited (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 403-419 Strongly oscillating nonhomogeneous Dirichlet condition (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 421-470 Some sharp error estimates (Doina Cioranescu, Alain Damlamian, Georges Griso)....Pages 471-492 Back Matter ....Pages 493-513