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Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations

  • Luís Balsa Bicho (a1) and António Ornelas (a1)

Abstract

We prove uniform continuity of radially symmetric vector minimizers uA(x) = UA(|x|) to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on a ball BR ⊂ ℝd, among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using a jointly convex lsc L∗∗ : ℝm×ℝ → [0,∞] with L∗∗(S,·) even and superlinear. Besides such basic hypotheses, L∗∗(·,·) is assumed to satisfy also a geometrical constraint, which we call quasi − scalar; the simplest example being the biradial case L∗∗(|u(x)|,|Du(x)|). Complete liberty is given for L∗∗(S,λ) to take the ∞ value, so that our minimization problem implicitly also represents e.g. distributed-parameter optimal control problems, on constrained domains, under PDEs or inclusions in explicit or implicit form. While generic radial functions u(x) = U(|x|) in this Sobolev space oscillate wildly as |x| → 0, our minimizing profile-curve UA(·) is, in contrast, absolutely continuous and tame, in the sense that its “static levelL∗∗(UA(r),0) always increases with r, a original feature of our result.

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[1] Bicho, L.B. and Ornelas, A., Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients. Nonlinear Anal. 74 (2011) 70617070.
[2] C. Carlota, S. Chá and A. Ornelas, Existence of radially increasing minimizers for nonconvex vectorial multiple integrals in the calculus of variations or optimal control, preprint.
[3] Cellina, A. and Perrotta, S., On minima of radially symmetric fuctionals of the gradient. Nonlinear Anal. 23 (1994) 239249.
[4] Cellina, A. and Vornicescu, M., On gradient flows. J. Differ. Eqs. 145 (1998) 489501.
[5] Crasta, G., Existence, uniqueness and qualitative properties of minima to radially-symmetric noncoercive nonconvex variational problems. Math. Z. 235 (2000) 569589.
[6] Crasta, G., On the minimum problem for a class of noncoercive nonconvex functionals. SIAM J. Control Optim. 38 (1999) 237253.
[7] Crasta, G. and Malusa, A., Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems. J. Convex Anal. 7 (2000) 167181.
[8] I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam (1976).
[9] Krömer, S., Existence and symmetry of minimizers for nonconvex radially symmetric variational problems. Calc. Var. PDEs 32 (2008) 219236.
[10] Krömer, S. and Kielhöfer, H., Radially symmetric critical points of non-convex functionals. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 12611280.
[11] P. Pedregal and A. Ornelas (editors), Mathematical methods in materials science and enginneering. CIM 1997 summerschool with courses by N. Kikuchi, D. Kinderlehrer, P. Pedregal. CIM www.cim.pt (1998).
[12] J. Yeh, Lectures on Real Analysis. World Scientific, Singapore (2006).

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