Matrix Weighted Poincaré Inequalities and Applications to Degenerate Elliptic Systems

We prove Poincaré and Sobolev inequalities inmatrix A p weighted spaces. We then use these Poincaré inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A p weight. Such results parallel earlier results by Fabes,...

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Bibliographic Details
Published in:Indiana University mathematics journal 2019-01, Vol.68 (5), p.1327-1377
Main Authors: Isralowitz, Joshua, Moen, Kabe
Format: Article
Language:English
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Summary:We prove Poincaré and Sobolev inequalities inmatrix A p weighted spaces. We then use these Poincaré inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A p weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A p weight. In addition, we prove Cacciopoli and reverse Meyers Hölder inequalities for weak solutions of the degenerate systems. Moreover, we show that the Riesz potential and fractional maximal operators are bounded on matrix weighted Lp spaces and go on to develop an entire matrix A p,q theory.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2019.68.7686