Continuous Steiner‐Symmetrization

where K is a convex subset of some functions space, e.g. W$j'(SZ) or Lp(sl), p > 1, and SZ c R" is a domain lying symmetrically to the hyperplane {y = 0}, {x = (x', y) , x' E R"-', y E R). If u E K , we often also have u* E K , where u* denotes the Steiner-symmetrization of u with respect to y , and J(u*) I J(u) . It can be proved for the absolute minimum u of (1) that u = u*. This argumentation fails for local minima or stationary points w of the functional J . Therefore the following question is natural: Is there a (in the norm of X ) continuous homotopy t H d , O I t < + 0 3 , u 0 = u , U r n = u * ,