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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate elliptic operators as regularizers

Author: R. N. Pederson
Journal: Trans. Amer. Math. Soc. 280 (1983), 533-553
MSC: Primary 35J70
MathSciNet review: 716836
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Abstract: The spaces $ {\mathcal{K}_{mk}}$, introduced in the Nehari Volume of Journal d'Analyse Mathématique, for nonnegative integer values of $ m$ and arbitrary real values of $ k$ are extended to negative values of $ m$. The extension is consistent with the equivalence $ \parallel {\zeta ^j}u{\parallel_{m,k}}\sim\parallel u{\parallel_{m,k - j}}$, the inequality $ \parallel {D^\alpha }u{\parallel_{m,k}} \leqslant {\text{const}}\parallel u{\parallel_{m + \vert\alpha \vert,k + \vert\alpha \vert}}$, and the generalized Cauchy-Schwarz inequality $ \vert\langle {u,v} \rangle \vert \leqslant \parallel u\,{\parallel_{m,k}}\parallel v\parallel_{ - m, - k}$. (Here $ \langle u, \upsilon \rangle$ is the $ {L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps $ {\mathcal{K}_{m,k}}\,1 - 1$ onto $ {\mathcal{K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)

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Article copyright: © Copyright 1983 American Mathematical Society