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

Luís Balsa Bicho; António Ornelas

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 1, page 141-157
- ISSN: 1292-8119

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topBicho, Luís Balsa, and Ornelas, António. "Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 141-157. <http://eudml.org/doc/272883>.

@article{Bicho2014,

abstract = {We prove uniformcontinuity of radiallysymmetric vector minimizers uA(x) = UA(|x|) to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on a ballBR ⊂ ℝd, among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using a jointlyconvexlscL∗∗ : 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 optimalcontrol problems, on constraineddomains, 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, absolutelycontinuous and tame, in the sense that its “staticlevel” L∗∗(UA(r),0) always increases with r, a original feature of our result.},

author = {Bicho, Luís Balsa, Ornelas, António},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {vectorial calculus of variations; vectorial distributed-parameter optimal control; continuous radially symmetric monotone minimizers},

language = {eng},

number = {1},

pages = {141-157},

publisher = {EDP-Sciences},

title = {Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations},

url = {http://eudml.org/doc/272883},

volume = {20},

year = {2014},

}

TY - JOUR

AU - Bicho, Luís Balsa

AU - Ornelas, António

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2014

PB - EDP-Sciences

VL - 20

IS - 1

SP - 141

EP - 157

AB - We prove uniformcontinuity of radiallysymmetric vector minimizers uA(x) = UA(|x|) to multiple integrals ∫BRL**(u(x), |Du(x)|) dx on a ballBR ⊂ ℝd, among the Sobolev functions u(·) in A+W01,1 (BR, ℝm), using a jointlyconvexlscL∗∗ : 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 optimalcontrol problems, on constraineddomains, 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, absolutelycontinuous and tame, in the sense that its “staticlevel” L∗∗(UA(r),0) always increases with r, a original feature of our result.

LA - eng

KW - vectorial calculus of variations; vectorial distributed-parameter optimal control; continuous radially symmetric monotone minimizers

UR - http://eudml.org/doc/272883

ER -

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