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Motivated by applications to problems from physics, we study elliptic operators with operator-valued coefficients acting on Banach-space-valued distributions. After giving a definition of ellipticity, normal ellipticity in particular, generalizing the classical concepts, we show that normally elliptic operators are negative generators of analytic semigroups on $ L_p({\Bbb R}^n, E) $ for 1 $ \leq p < \infty $ and on $ BUC({\Bbb R}^n, E) $ and $ C_0({\Bbb R}^n, E) $ , as well as on all Besov spaces of E-valued distributions on $ {\Bbb R}^n $ , where E is any Banach space. This is true under minimal regularity assumptions for the coefficients, thanks to a point-wise multiplier theorem for E-valued distributions proven in the appendix.
Journal of Evolution Equations – Springer Journals
Published: Jun 1, 2001
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