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Lower semicontinuity in domain optimization problems

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In the present paper, the lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by Chenais is employed. For this class of domains, a basic lemma is proved that plays an essential role in the derivations of the lower-semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.

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The author wishes to express his sincere thanks to the reviewer for his valuable comments, which made the paper more readable; the reviewer also pointed out that Lemma 2.1 in the text is a direct corollary to a lemma by Chenais (Ref. 9). He thanks Prof. Y. Sakawa of Osaka University for encouragement.

Communicated by E. J. Haug, Jr.

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Fujii, N. Lower semicontinuity in domain optimization problems. J Optim Theory Appl 59, 407–422 (1988). https://doi.org/10.1007/BF00940307

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Key Words

  • Lower semicontinuity
  • domain optimization
  • boundary-value problems
  • Dirichlet problems
  • Neumann problems