Constructions of bounded solutions of div\, u=f in critical spaces

We construct uniformly bounded solutions of the equation div\, u=f for arbitrary data f in the critical spaces Lᵈ(Ω), where Ω is a domain of Rᵈ. This question was addressed by Bourgain & Brezis, [On the equation div\, Y=f and application to control of phases, JAMS 16(2) (2003) 393-426], who proved that although the problem has a uniformly bounded solution, it is critical in the sense that there exists no linear solution operator for general Lᵈ-data. We first discuss the validity of this existence result under weaker conditions than f∈ Lᵈ(Ω), and then focus our work on constructive processes for such uniformly bounded solutions. In the $d=2$ case, we present a direct one-step explicit construction, which generalizes for $d>2$ to a $(d-1)$-step construction based on induction. An explicit construction is proposed for compactly supported data in L2,∞(Ω) in the $d=2$ case. We also present constructive approaches based on optimization of a certain loss functional adapted to the problem. This approach provides a two-step construction in the $d=2$ case. This optimization is used as the building block of a hierarchical multistep process introduced in [E. Tadmor, Hierarchical construction of bounded solutions in critical regularity spaces, CPAM 69(6) (2016) 1087-1109] that converges to a solution in more general situations.