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Elliptic Operators with In nite-Dimensional State Spaces Herbert Amann

Summary: Elliptic Operators with In nite-Dimensional State Spaces
Herbert Amann
Abstract. Motivated by applications to problems from physics, we study elliptic operators with
operator-valued coeÆcients acting on Banach-space-valued distributions. After giving a de ni-
tion of ellipticity, normal ellipticity in particular, generalizing the classical concepts, we show
that normally elliptic operators are negative generators of analytic semigroups on Lp (R n ; E) for
1  p < 1, and on BUC(R n ; E) and C 0 (R n ; E), as well as on all Besov spaces of E-valued distri-
butions on R n , where E is any Banach space. This is true under minimal regularity assumptions
for the coeÆcients, thanks to a point-wise multiplier theorem for E-valued distributions proven
in the appendix.
Mathematical Subject Classi cation (1991). 35J30, 35J45, 35K25, 35S50, 46F25, 47D06,
47G99 .
Keywords. Elliptic operators with operator-valued coeÆcients, resolvent estimates, analytic
semigroups, vector-valued Besov spaces, Lebesgue spaces, and spaces of continuous and Holder
continuous functions, point-wise multipliers for vector-valued Besov spaces.
In this paper we derive resolvent estimates for linear elliptic di erential operators
A := A(x; D) :=
j jm


Source: Amann, Herbert - Institut für Mathematik, Universität Zürich


Collections: Mathematics