Mathematics – Analysis of PDEs
Scientific paper
2010-03-07
Mathematics
Analysis of PDEs
Scientific paper
We construct uniformly bounded solutions for the equations div U=f and curl U= f in the critical cases f \in L^d_#(T^d,R) and f\in L^3_#(R^3,R^3). Bourgain & Brezis, \cite{BB03,BB07}, have shown that there exists no \emph{linear} construction for such solutions. Our constructions are special cases of a general framework for solving linear equations of the form T U=f, where T is a linear operator densely defined in Banach space B with a closed range in a (proper subspace) of Lebesgue space L^p_#(\Omega), and with an injective dual T^*. The solutions are realized in terms of a multiscale \emph{hierarchical representation}, U=\sum_{j=1}^\infty u_j, interesting for its own sake. Here, the u_j's are constructed \emph{recursively} as minimizers of the iterative refinement step, u_{j+1} = {\arginf}_{u}\big\{\|u\|_B+\lambda_{j+1}\|r_j-T u \|^p_{\L^p(\Omega)}\big\}, where r_j:=f-T(\sum^j_{k=1}u_k), are resolved in terms of an exponentially increasing sequence of scales \lambda_{j+1}:=\lambda_1 \zeta^j. The resulting hierarchical decompositions, U =\sum_{j=1}^\infty u_j, are nonlinear.
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