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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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