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dc.contributor.authorLopes, Orlando ( )
dc.identifier.citationLopes, O. (1996). Radial and nonradial minimizers for some radially symmetric functionals. Electronic Journal of Differential Equations, 1996(03), pp. 1-14.en_US

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 ΣKi=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 continua- tion principle for elliptic systems plays a crucial role in our method.

dc.format.extent14 pages
dc.format.medium1 file (.pdf)
dc.publisherSouthwest Texas State University, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 1996, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectVariational problemsen_US
dc.subjectRadial and nonradial minimizersen_US
dc.titleRadial and Nonradial Minimizers for Some Radially Symmetric Functionalsen_US
dc.rights.licenseCreative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.



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