We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

中文翻译：

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Lipschitz界和非均匀椭圆率

我们考虑非均匀椭圆变分问题，并根据给定数据的正则性给出最优条件，以保证解的局部Lipschitz正则性。分析抓住了文献中的主要模型案例。涵盖了具有快速，指数型增长条件的积分以及具有不平衡多项式增长条件的积分。我们的标准涉及自然极限函数空间，并在这种非常笼统的背景下，再现了泊松方程线性情况下已知的经典和最优函数空间。此外，我们提供了新的自然增长的先验估计，其有效性是一个未解决的问题。最后，我们在经典一致椭圆型情况下也发现了新结果。除了特定的结果外，本文还提出了一种解决不均匀椭圆率的新方法，在某种意义上，允许我们通过潜在的理论论点将非均匀椭圆问题减少为均匀椭圆问题，这是首次在这种情况下应用的。©2019作者。《纯数学和应用数学通讯》由Courant数学科学学院和Wiley Periodicals，Inc.发布。