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Regularity results for minimizers of irregular integrals with \((p,q)\) growth. (English) Zbl 0999.49022

The authors study conditions that have to be imposed on the growth of function \(f\) to guarantee local Lipschitz-continuity of the solution to the minimization problem \[ \mathcal{F} = \int_\Omega f(Du) dx . \] Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\), \(n\geq 2\). The admissible functions \(u\) are defined in \(\Omega\) and take the values in \(\mathbb{R}^N\). The integrand function \(f: \mathbb{R}^{Nn}\rightarrow \mathbb{R}\) is nonnegative and convex. Three cases are considered: \(N=1\) (scalar admissible functions \(u\)), \(f(z)=g(|z|)\) (spherical symmetry), and the general case. In each situation the growth and strong convexity conditions of the function \(f\) are imposed, so that the minimizing function \(u\) is locally Lipschitz-continuous. The growth conditions on all cases are formulated as double inequality \[ \left(\mu^2+|z|^2\right)^{p/2}\leq f(z) \leq L\left(\mu^2+|z|^2\right)^{p/2} + L \left(\mu^2+|z|^2\right)^{q/2} \] where \(L\geq 1\), \(\mu\in[0,1]\), \(p\) and \(q\) are constants such that \(1<p\leq q\), \(q/p<1+1/n\). The strong convexity requirement is formulated differently in each case. For example, in the general case the inequality \[ \theta f(z_1) +(1-\theta)f(z_2)-f(\theta z_1+(1-\theta)z_2) \geq 1/2\nu \theta(1-\theta)(\mu^2+|z_1|^2+|z_2|^2)^{p/2-1}\left|z_1-z_2\right|^2 \] should hold for some constant \(\nu\in(0,1]\) and arbitrary \(z_1\), \(z_2\) and \(\theta\in(0,1)\).
The operator \(D\) is not specified in the paper. Apparently it is a first order differential operator. The conditions obtained in the paper do not require \(Du\) to be defined at every point of \(\Omega\).
No example of application of the obtained results is provided.
Reviewer: D.Silin (Berkeley)

MSC:

49N60 Regularity of solutions in optimal control
35J20 Variational methods for second-order elliptic equations
49J10 Existence theories for free problems in two or more independent variables
49N10 Linear-quadratic optimal control problems
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