# Trace operators of the bi-Laplacian and applications

## TL;DR

This paper develops and analyzes trace operators related to the bi-Laplacian for ultraweak formulations of the bi-Laplace equation, enabling low-regularity solutions and demonstrating their effectiveness through numerical experiments.

## Contribution

It introduces new trace operators and formulations for the bi-Laplacian that are well-posed under low regularity, extending to higher dimensions.

## Key findings

- Two well-posed ultraweak formulations are proposed.
- Numerical experiments confirm the effectiveness of the formulations.
- Analysis applies to dimensions two and higher.

## Abstract

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 non-closed space. This can be fixed by switching to the Kirchhoff-Love traces from [F\"uhrer, Heuer, Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp., 88 (2019)]. Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the DPG 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 non-closedness of a trace space), our analysis applies to any space dimension larger than or equal to two.