## 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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