Constructions of Bounded Solutions of \( \textit{di}\upsilon \) u = f in Critical Spaces

We construct uniformly bounded solutions of the equation \( \textit{di}\upsilon \) u = f for arbitrary data f in the critical spaces \( \textit{L}^{d}(\Omega) \) , where Ω is a domain of \( \mathbb{R}^{\textit{d}} \) . This question was addressed by Bourgain & Brezis, [BB2003], 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 \( \textit{L}^{d} \) -data. We first discuss the validity of this existence result under weaker conditions than \( \textit{f } \epsilon \textit{L}^{\textit{d}} \) , 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 also proposed for compactly supported data in \( \textit{L}^{\textit{d},\infty} \) . We finally 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 [Tad2014] that converges to a solution in more general situations.