We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity under the condition of an |$L_2$| right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a nonclosed space. This can be fixed by switching to the Kirchhoff–Love traces from Führer et al. (2019, An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation. Math. Comp., 88, 1587–1619). Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the discontinuous Petrov–Galerkin method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the nonclosedness of a trace space) our analysis applies to any space dimension larger than or equal to two.

中文翻译：

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双Laplacian的跟踪运算符及其应用

我们研究了几个与双Laplacian有关的跟踪算子和空间。它们是由具有均匀Dirichlet条件的bi-Laplace方程的超弱公式的开发所激发的，但也与描述混合逼近的一致性有关。我们的目标是在| $ L_2 $ |的条件下，提出具有良好正则性（超弱）的公式，这些公式假定低规则性。右侧功能。我们采用两种定义迹线的方法以及相应的零件积分公式。在一种情况下，人们获得了一个非封闭空间。可以通过切换到Führer等人的Kirchhoff-Love轨迹来解决此问题。（2019，Kirchhoff–Love板弯曲模型和DPG近似的超弱公式。，88，1587–1619）。使用跟踪运算符的不同组合，我们获得了两个恰当的公式。对于这两种方法，我们都报告了用不连续的Petrov-Galerkin方法和最佳测试函数进行的数值实验。在本文中，我们考虑了两个和三个空间维度。但是，除了附录中的给定反例（与跟踪空间的非封闭性有关）外，我们的分析适用于任何大于或等于2的空间维。