Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces |
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Authors: | Eitan Tadmor |
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Institution: | Department of Mathematics Center of Scientific Computation and Mathematical Modeling and Institute for Physical Science and Technology, University of Maryland, College Park, MD |
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Abstract: | We construct uniformly bounded solutions for the equations div U = f and U = f in the critical cases and , respectively. Criticality in this context manifests itself by the lack of a linear solution operator mapping . Thus, the intriguing aspect here is that although the problems are linear, construction of their solutions is not. Our constructions are special cases of a general framework for solving linear equations of the form , where is a linear operator densely defined in Banach space with a closed range in a (proper subspace) of Lebesgue space , and with an injective dual . The solutions are realized in terms of a multiscale hierarchical representation, , interesting for its own sake. Here, u j's are constructed recursively as minimizers of where the residuals are resolved in terms of a dyadic sequence of scales with large enough . The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve in critical regularity spaces.© 2016 Wiley Periodicals, Inc. |
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