Existence and relaxation problems in optimal shape design

Zdzisław Denkowski

DOI: http://dx.doi.org/10.12775/TMNA.2000.036


A general abstract theorem on existence of solutions to optimal shape
design problems for systems governed by partial differential equations,
or variational
inequalities or hemivariational inequalities is formulated and two main
properties (conditions) responsible for the existence are discussed.
When one of them fails one have to make ``relaxation'' in order to get some generalized
optimal shapes. In particular, some relaxation ``in state'', based on $\Gamma$
convergence, is presented in details for elliptic, parabolic and hyperbolic
PDEs (and then for optimal shape design problems), while the relaxation
``in cost functional'' is discussed for some special classes of functionals.


Optimal shape design; existence; optimal solutions; relaxation

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