Radial and Nonradial Minimizers for Some Radially Symmetric Functionals
Date
1996-02-29
Authors
Lopes, Orlando
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
In a previous paper we have considered the functional
V(u) = 1\2 ∫ℝN | grad u(x)|2 dx + ∫ℝN F(u(x))dx
subject to
∫ℝN G(u(x)) dx = λ > 0,
where u(x) = (u1(x),..., uK(x)) belongs to H1K(ℝN) = H1(ℝN) x ∙∙∙ x H1(ℝN) (K times) and | grad u(x)|2 means ∑K i=1 | grad ui(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 Ω which is invariant under rotations but not under translations, that is, Ω 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.
Description
Keywords
Variational problems, Radial and nonradial minimizers
Citation
Lopes, O. (1996). Radial and nonradial minimizers for some radially symmetric functionals. <i>Electronic Journal of Differential Equations, 1996</i>(03), pp. 1-14.
Rights
Attribution 4.0 International
Rights Holder
This work is licensed under a Creative Commons Attribution 4.0 International License.