### Summary

Summary: In a previous paper we have considered the functional $$V(u) = {1\over 2}\int_{ R^N} |{\rm\ grad}\, u(x)|^2\, dx + \int_{ R^N}F(u(x))\,dx$$ subject to $\int_{ R^N} G(u(x))\, dx = \lambda$ where $u(x) = (u_1(x) , \ldots, u_K(x))$ belongs to $H^1_K (R^N) = H^1 (R^N) \times\cdots\times H^1( R^N)$ (K times) and $|{\rm\ grad}\, u(x)|^2$ means $\sum^K_{i=1}|{\rm\ grad}\, u_i (x)|^2$. We have shown that, under some technical assumptions and except for a translation in the space variable x, any global minimizer is radially symmetric. In this paper we consider a similar question except that the integrals in the definition of the functionals are taken on some set $\Omega$ which is invariant under rotations but not under translations, that is, $\Omega$ is either a ball, an annulus or the exterior of a ball. In this case we show that for the minimization problem without constraint, global minimizers are radially symmetric. However, for the constrained problem, in general, the minimizers are not radially symmetric. For instance, in the case of Neumann boundary conditions, even local minimizers are not radially symmetric (unless they are constant). In any case, we show that the global minimizers have a symmetry of codimension at most one. We use our method to extend a very well known result of Casten and Holland to the case of gradient parabolic systems. The unique continuation principle for elliptic systems plays a crucial role in our method.

35J20, 49J10