Shape controllability for free boundaries. (English) Zbl 0543.49016

Zolésio, J. P.

System modelling and optimization, Proc. 11th IFIP Conf., Copenhagen 1983, Lect. Notes Control Inf. Sci. 59, 354-361 (1984).

Summary: [For the entire collection see Zbl 0538.00033.]
We study the evolution of the free boundary \(\Sigma\) arising in the classical obstacle problem. When the boundary \(\Gamma\) of the domain is perturbed by a vector field V we give a representation of the vector field W which builds the deformation of \(\Sigma\).
In the classical obstacle problem for the membrane the free boundary appears as the zero level set of the function \(Y=y-\psi\) (where y is the displacement of the membrane and \(\psi\) the obstacle, the constraint being \(y\geq \psi).\)
This situation is general in many problems. We first study the evolution of a level curve when some parameters, may be the geometry, have given variations. As an introduction we recall some previous results. Then, given a speed deformation V of the space \(\Omega\) of the membrane, we investigate the speed deformation W of the free boundary \(\Sigma\).

MSC:

93B03	Attainable sets, reachability
35R35	Free boundary problems for PDEs
49J40	Variational inequalities
35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)
74K15	Membranes

Keywords:

free boundary; obstacle problem; membrane; deformation

Citations:

Zbl 0538.00033