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Nondiffusive variational problems with distributional and weak gradient constraints. (English) Zbl 1498.35293

Summary: 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.

MSC:

35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
90C46 Optimality conditions and duality in mathematical programming
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