This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.

中文翻译：

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一个受完美塑性极限分析启发的抽象 inf-sup 问题及相关应用

本文关注由双线性拉格朗日和凸约束生成的抽象 inf-sup 问题。我们研究了保证 inf-sup 和相关的 sup-inf 问题之间没有差距的条件。论文中引入的关键假设概括了众所周知的 Babuška-Brezzi 条件。它基于为函数空间中的凸锥定义的 inf-sup 条件。我们还应用了一种便于解决 inf-sup 问题的正则化方法，并推导出临界 (inf-sup) 值的可计算大数，可用于后验的数值结果的误差分析。针对抽象问题获得的结果应用于连续介质力学。特别介绍了在经典塑性、梯度塑性和分层中出现的极限载荷问题和类似问题的例子。