Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 141-157.

We prove uniform continuity of radially symmetric vector minimizers u A ( x ) = U A ( | x | ) to multiple integrals B R L * * ( u ( x ) , | D u ( x ) | ) d x on a ball B R d , among the Sobolev functions u ( · ) in A + W 0 1 , 1 ( B R , m ) , using a jointly convex lsc L * * : m × [ 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 ) | , | D u ( 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 U A ( · ) is, in contrast, absolutely continuous and tame, in the sense that its “static level L * * ( U A ( r ) , 0 ) always increases with r, a original feature of our result.

DOI : 10.1051/cocv/2013058
Classification : 49J10, 49N60
Mots clés : vectorial calculus of variations, vectorial distributed-parameter optimal control, continuous radially symmetric monotone minimizers
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     title = {Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations},
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     pages = {141--157},
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Bicho, Luís Balsa; Ornelas, António. Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 141-157. doi : 10.1051/cocv/2013058. http://www.numdam.org/articles/10.1051/cocv/2013058/

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