Higher integrability for variational integrals with non-standard growth. (English) Zbl 1462.49015
Summary: We consider autonomous integral functionals of the form
\[
\mathcal{F} [u]:=\int_\Omega f(Du)\,dx \text{ where } u:\Omega \rightarrow \mathbb{R}^N,\, N\geq 1,
\]
where the convex integrand \(f\) satisfies controlled \((p,q)\)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of \(\mathcal{F}\) assuming \(\frac{q}{p}<1+\frac{2}{n-1}\), \(n\geq 3\). This improves earlier results valid under the more restrictive assumption \(\frac{q}{p}<1+\frac{2}{n}\).
MSC:
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 49N60 | Regularity of solutions in optimal control |
| 35J70 | Degenerate elliptic equations |
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