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

Bicho, Luís Balsa; Ornelas, António

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 141-157.

We prove uniform continuity of radially symmetric vector minimizers $u_A\left(x\right)=U_A\left(\right|x\left|\right)$ to multiple integrals ${\int }_{B_R}{L}^{**}(u\left(x\right),|Du\left(x\right)\left|\right)dx$ on a ball $B_R\subset {ℝ}^d$, among the Sobolev functions $u(·)$ in $A+W_0^{1,1}(B_R,{ℝ}^m)$, using a jointly convex lsc $L^{**}:{ℝ}^m\times ℝ\to [0,\infty ]$ 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 $L^{**}\left(\right|u\left(x\right)|,|Du\left(x\right)\left|\right)$. Complete liberty is given for $L^{**}(S,\lambda )$ to take the $\infty$ 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 $u\left(x\right)=U\left(\right|x\left|\right)$ in this Sobolev space oscillate wildly as $\left|x\right|\to 0$, our minimizing profile-curve $U_A(·)$ is, in contrast, absolutely continuous and tame, in the sense that its “static level” $L^{**}(U_A\left(r\right),0)$ always increases with $r$, a original feature of our result.

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