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Weak shape formulation of free boundary problems. (English) Zbl 0807.49018
Several classical free boundary (fb) problems can be formulated as minimization problems which can be understood as variational inequalities. In this context the weak shape formulation was introduced earlier by the author [e.g., Optimization of distributed parameter structures, Vol. II, NATO Adv. Study Inst., Ser. E Appl. Sci. 50, 1152- 1194 (1981; Zbl 0537.35074)]. This formulation is not equivalent to a variational inequality and permits to handle more general fb conditions. Five classical fb situations related to scalar elliptic problems and the Bernoulli fb condition are discussed in this paper. The first two deal with the Dirichlet condition on the boundary of the domain; they correspond to the fb condition associated to a perfect 2D fluid (using a stream function representation). The fourth one corresponds to the Neumann condition, which can be related to the 3D perfect fluid (using a potential formulation). The last one corresponds to transmission conditions through the boundary, while the third one is a mixed situation.
Reviewer: V.Arnautu (Iaşi)

MSC:
49K10 Optimality conditions for free problems in two or more independent variables
49J20 Existence theories for optimal control problems involving partial differential equations
35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
49Q10 Optimization of shapes other than minimal surfaces
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