论文标题
占用量的清晰度和非偏差度衡量整体变化问题的范围
Sharpness and non-sharpness of occupation measure bounds for integral variational problems
论文作者
论文摘要
我们分析了最近提出的两种方法，以建立对一般整体变化问题的最小值的先验下限。这些方法涉及“职业措施”或“双重双重弛豫”程序，显示出在强制性假设下产生相同的下限，从而确保其强二重性。然后，我们通过minimax参数表明，该方法实际上评估了一维，标量值或凸多维问题类别的最小值。但是，对于通用问题，这些方法应无法捕获最小值并产生非偏差下限。我们使用两个示例证明了这一点，第一个示例是一维且标量为非convex约束的标量，第二个是多维和非凸的，以不同的方式。后一个示例强调了非线性约束在梯度领域的多个维度的存在，这些梯度措施忽略了梯度措施，但内置在更精细的渐变年轻措施理论中。
We analyze two recently proposed methods to establish a priori lower bounds on the minimum of general integral variational problems. The methods, which involve either `occupation measures' or a `pointwise dual relaxation' procedure, are shown to produce the same lower bound under a coercivity hypothesis ensuring their strong duality. We then show by a minimax argument that the methods actually evaluate the minimum for classes of one-dimensional, scalar-valued, or convex multidimensional problems. For generic problems, however, these methods should fail to capture the minimum and produce non-sharp lower bounds. We demonstrate this using two examples, the first of which is one-dimensional and scalar-valued with a non-convex constraint, and the second of which is multidimensional and non-convex in a different way. The latter example emphasizes the existence in multiple dimensions of nonlinear constraints on gradient fields that are ignored by occupation measures, but are built into the finer theory of gradient Young measures.
