Abstract Integrals of the Calculus of Variations with p , q {p,q} -growth may have not smooth minimizers, not even bounded, for general p , q {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f ⁢ ( x , ξ ) {f(x,\xi)} with dependence on the modulus of the gradient, i.e. f ⁢ ( x , ξ ) = g ⁢ ( x , | ξ | ) {f(x,\xi)=g(x,|\xi|)} . Without imposing structure conditions, we prove that if q p {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

中文翻译：

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无结构条件的标量积分的正则性

对于一般的 p , q {p,q} 指数，具有 p , q {p,q} -growth 的变分积分的抽象积分可能没有平滑的极小值，甚至没有界。在本文中，我们考虑标量情况，与矢量值相反，它允许我们不将结构条件强加给被积函数 f ⁢ ( x , ξ ) {f(x,\xi)} 依赖于梯度，即 f ⁢ ( x , ξ ) = g ⁢ ( x , | ξ | ) {f(x,\xi)=g(x,|\xi|)} 。在不施加结构条件的情况下，我们证明如果 qp {\frac{q}{p}} 足够接近 1，那么每个极小值都是局部 Lipschitz 连续的。