Calculating tangent sets to certain sets in functional spaces

Ewa Bednarczuk (1) , Michel Pierre , Elisabeth Rouy , Jan Sokolowski (1)

1 NUMATH - Mathematical Analysis and Numerical Simulation of Non-Linear Models

We give necessary and sufficient conditions for a given element to be a member of the second order tangent set $T»_{K}(f,v)$ to the positive cone $K$ in $L^{\infty}¸.$ Since, in general $T»_{K}(f,v)$ may be empty we give conditions on functions $f¸, v$ which ensure that the second tangent set is a cone. As an application of the results obtained we give a characterization of the elements of the first and second tangent set to the set $B={u\in W^{1,\infty}(Ømega)¸ |¸|\nabla u|^{2}\leq 1}¸.$

Keywords: tangent cone; second order tangent sets; shape optimization

Domains: Other [cs.OH]

Ewa Bednarczuk, Michel Pierre, Elisabeth Rouy, Jan Sokolowski. Calculating tangent sets to certain sets in functional spaces. [Research Report] RR-3190, INRIA. 1997, pp.24. ⟨inria-00073499⟩