Abstract The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation f ∈ L p , p > 1 . If f is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to L 2 .

中文翻译：

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拉格朗日乘子和非常量梯度约束问题

摘要 本文的目的是研究与线性算子相关的梯度约束问题。研究了两类问题。第一个是非常量梯度约束问题和合适的障碍物问题之间的等价性，其中障碍物解决粘性意义上的 Hamilton-Jacobi 方程。等价结果是在梯度约束条件下得到的。第二个问题是拉格朗日乘子的存在。我们证明了非常量梯度约束问题允许一个拉格朗日乘子，如果方程 f ∈ L p , p > 1 的自由项，它是一个氡测度。如果 f 是一个正常数，我们将结果正则化，即我们证明拉格朗日乘子属于 L 2 。