## 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.

### MSC:

 49N60 Regularity of solutions in optimal control
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### References:

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