We consider autonomous integral functionals of the form

$$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$

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 \(\frac{q}{p}<1+\frac{2}{n-1}\), \(n\ge 3\). This improves earlier results valid under the more restrictive assumption \(\frac{q}{p}<1+\frac{2}{n}\).

中文翻译：

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非标准增长的变分积分具有更高的可积性

我们考虑以下形式的自治积分函数

$$ \ begin {aligned} {\ mathcal {F}} [u]：= \ int _ \ varOmega f（D u）\，dx \ quad \ text {其中} u：\ varOmega \ rightarrow {\ mathbb {R }} ^ N，N \ ge 1，\ end {aligned} $$

凸被积物f满足控制的（p，  q）-增长条件。我们建立的极小更高梯度积和部分规律性\（{\ mathcal {F}} \）假设\（\压裂{Q} {P} <1+ \压裂{2} {N-1} \），\ （n \ ge 3 \）。这会改善在更严格的假设\（\ frac {q} {p} <1+ \ frac {2} {n} \）下有效的早期结果。