G-Convergence and Homogenization of Nonlinear Partial Differential Operators:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
1997
|
| Series: | Mathematics and Its Applications
422 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space |
| Physical Description: | 1 Online-Ressource (XIII, 258 p) |
| ISBN: | 9789401589574 9789048149001 |
| DOI: | 10.1007/978-94-015-8957-4 |
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| 500 | |a Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space | ||
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| institution | BVB |
| isbn | 9789401589574 9789048149001 |
| language | English |
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| spelling | Pankov, Alexander Verfasser aut G-Convergence and Homogenization of Nonlinear Partial Differential Operators by Alexander Pankov Dordrecht Springer Netherlands 1997 1 Online-Ressource (XIII, 258 p) txt rdacontent c rdamedia cr rdacarrier Mathematics and Its Applications 422 Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space Mathematics Differential equations, partial Partial Differential Equations Mathematik Nichtlinearer partieller Differentialoperator (DE-588)4171764-8 gnd rswk-swf Homogenisierung Mathematik (DE-588)4403079-4 gnd rswk-swf Konvergenz (DE-588)4032326-2 gnd rswk-swf Nichtlinearer partieller Differentialoperator (DE-588)4171764-8 s Konvergenz (DE-588)4032326-2 s 1\p DE-604 Homogenisierung Mathematik (DE-588)4403079-4 s 2\p DE-604 https://doi.org/10.1007/978-94-015-8957-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Pankov, Alexander G-Convergence and Homogenization of Nonlinear Partial Differential Operators Mathematics Differential equations, partial Partial Differential Equations Mathematik Nichtlinearer partieller Differentialoperator (DE-588)4171764-8 gnd Homogenisierung Mathematik (DE-588)4403079-4 gnd Konvergenz (DE-588)4032326-2 gnd |
| subject_GND | (DE-588)4171764-8 (DE-588)4403079-4 (DE-588)4032326-2 |
| title | G-Convergence and Homogenization of Nonlinear Partial Differential Operators |
| title_auth | G-Convergence and Homogenization of Nonlinear Partial Differential Operators |
| title_exact_search | G-Convergence and Homogenization of Nonlinear Partial Differential Operators |
| title_full | G-Convergence and Homogenization of Nonlinear Partial Differential Operators by Alexander Pankov |
| title_fullStr | G-Convergence and Homogenization of Nonlinear Partial Differential Operators by Alexander Pankov |
| title_full_unstemmed | G-Convergence and Homogenization of Nonlinear Partial Differential Operators by Alexander Pankov |
| title_short | G-Convergence and Homogenization of Nonlinear Partial Differential Operators |
| title_sort | g convergence and homogenization of nonlinear partial differential operators |
| topic | Mathematics Differential equations, partial Partial Differential Equations Mathematik Nichtlinearer partieller Differentialoperator (DE-588)4171764-8 gnd Homogenisierung Mathematik (DE-588)4403079-4 gnd Konvergenz (DE-588)4032326-2 gnd |
| topic_facet | Mathematics Differential equations, partial Partial Differential Equations Mathematik Nichtlinearer partieller Differentialoperator Homogenisierung Mathematik Konvergenz |
| url | https://doi.org/10.1007/978-94-015-8957-4 |
| work_keys_str_mv | AT pankovalexander gconvergenceandhomogenizationofnonlinearpartialdifferentialoperators |