##
**Regularity for some scalar variational problems under general growth conditions.**
*(English)*
Zbl 0901.49030

The author continues his study of regularity for minima of integral functionals of the form
\[
F(v) = \int_\Omega f(Dv) dx,
\]
where \(\Omega\) is an open set of the Euclidean \(n\)-space with \(n \geq 2\), and \(f\) is a scalar valued function. It is well known that if \(f\) behaves likes \(f_p\) defined by \(f^p(\xi) = | \xi| ^p\), then minimizers of \(F\) (subject to suitable boundary conditions) are at least locally Lipschitz. Here, the interest is in a more general class of functions \(f\). Specifically, the function \(f\) is assumed to be twice differentiable and the eigenvalues of the second derivative matrix \(\partial_2 f/\partial \xi^2\) are assumed to lie between two positive, increasing functions \(g_1(| \xi|)\) and \(g_2(| \xi|)\). In addition, the functions \(g_1\) and \(g_2\) are related, first, by the obvious inequality \(g_1 \leq g_2\), and by the restrictions
\[
g_2(t)t^2 = O \left(\int_0^{t} (g_1(s))^{1/2}) ds)^\alpha\right)
\]
and
\[
g_2(| \xi|)| \xi| ^2 = O \left( f(\xi)^\beta\right)
\]
as \(t\) and \(| \xi| \) go to \(\infty\) for exponents \(\alpha\) and \(\beta\) satisfying the inequalities
\[
2 \leq \sigma < \frac {2n}{n-2},\qquad 1 \leq\beta < \frac 2n \frac{\alpha}{\alpha-2}
\]
using the convention that \(a/0 =\infty\) if \(a > 0\). The main result is that the \(L^\infty(B_\rho)\) norm of the gradient of a minimizer can be estimated in terms of the \(L^1(B_r)\) norm of that gradient, where \(B_\rho\) and \(B_r\) are concentric balls with \(\rho < r\). After discussing a number of examples of functions \(f\) which satisfy these conditions, the author proves the gradient estimate via an approximation argument and a variant of the Moser iteration scheme.

Reviewer: G.M.Lieberman (Ames)

### MSC:

49N60 | Regularity of solutions in optimal control |

### Keywords:

calculus of variations; direct methods; regularity; nonstandard growth conditions; general growth conditions
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\textit{P. Marcellini}, J. Optim. Theory Appl. 90, No. 1, 161--181 (1996; Zbl 0901.49030)

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### References:

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