In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a nonstandard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solution to this pre-dual problem under some assumptions. We conclude the article by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples illustrate the theoretical findings.

中文翻译：

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具有分布和弱梯度约束的非扩散变分问题

在本文中，我们考虑具有混合边界条件和（分布和弱）梯度约束的非扩散变分问题。约束的上限是函数或 Borel 测度，导致状态空间是 Sobolev 或有界变化函数空间。我们在低规律性假设下解决模型的存在性和唯一性，并严格识别其 Fenchel 预对偶问题。在某些情况下，后者位于具有平方可积散度的 Borel 度量的非标准空间上。我们还在一些假设下建立了这个前对偶问题解的存在性和唯一性。我们通过引入混合有限元方法来解决原始对偶系统来结束本文。数值例子说明了理论发现。