Electronic Journal of Differential Equations,
Vol. 1996(1996), No. 3, pp 1--14.
Title: Radial and Nonradial Minimizers for Some Radially
Symmetric Functionals
Authors: Orlando Lopes (IMECC-UNICAMP, Brazil)
Abstract: 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
$$ \int_{R^N} G(u(x))\, dx = \lambda > 0,$$
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.
Submitted July 25, 1996. Published November 22, 1996.
Math Subject Classification: 35J20, 49J10.
Key Words: Variational Problems; Radial and Nonradial Minimizers.