Local Minimizers Induced by Spatial Inhomogeneity with Inner Transition Layer
✍ Scribed by Arnaldo Simal do Nascimento
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 906 KB
- Volume
- 133
- Category
- Article
- ISSN
- 0022-0396
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✦ Synopsis
The main aim of this paper is to construct and provide the structure of local minimizers of the variational problem inf v # L 1 (0) F = (v), where =: small parameter and
, otherwise having only the functions k 1 >0 and k 2 >0 as parameters. Given any closed simple smooth curve # m contained in a level set of a(X)=k 1 (X) k 2 (X), X # 0, we give necessary conditions relating the slope and concavity of a(X), X # # m , along the normal n^[# m (X)], with the curvature }(X) of # m so that the above problem possesses a nonconstant minimizer v = (X) whose nodal curve # = m converges uniformly to # m , as = Ä 0. Moreover v = Ä 1, as = Ä 0, uniformly on compact sets contained in one of the two connected components of 0"# m and v = Ä &1, as = Ä 0, uniformly on compact sets contained in the other one. 1997 Academic Press 1
where k 1 and k 2 are positive functions in C 2 (0 , R), 0: an open bounded subset of R 2 with a Lipschitz-continuous boundary.