Hierarchical Construction of Bounded Solutions of div(U)=f in Critical Regularity Spaces
Thursday, October 23, 2014 - 9:00am - 9:45am
We construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity spaces L^d(R^d). The study of this equation and related problems such as curl(U)=f in critical L^3(R^3), was motivated by recent results of Bourgain & Brezis. Although the problems are linear, the construction of their solutions is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U=∑u_j which we introduced earlier in the context of image processing. The u_j's are constructed recursively as proper minimizers, yielding a multi-scale decomposition of the solutions U.