Hierarchical solutions for linear equations: a constructive proof of the closed range theorem
Eitan Tadmor · arXiv (Cornell University) · 2010
ABSTRACT. We construct uniformly bounded solutions for the equations divU = f and curlU = f in the critical cases f ∈ Ld #(Td,R) and f∈L 3 #(R3,R 3). Bourgain & Brezis, [BB03, BB07], have shown that there exists no linear construction for such solutions. Our constructions are special cases of a general framework for solving linear equations of the form L U = f, where L is a linear operator densely defined in Banach spaceBwith a closed range in a (proper subspace) of Lebesgue space L p #(Ω), and with an injective dual L ∗. The solutions are realized in terms of a multiscale hierarchical representation, U = ∑ ∞ j=1 u j, interesting for its own sake. Here, the u j’s are constructed recursively as minimizers of the iterative refinement step, u j+1 = arginfu ‖u‖B+ λ j+1‖r j−L u ‖ p