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Transactions of the American Mathematical Society

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



An energy inequality for higher order linear parabolic operators and its applications

Author: David Ellis
Journal: Trans. Amer. Math. Soc. 165 (1972), 167-206
MSC: Primary 47G05; Secondary 35S05
MathSciNet review: 0298482
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Abstract: A generalization of the classical energy inequality is obtained for evolution operators $ (\partial /\partial t)I - H(t){\Lambda ^{2k}} - J(t)$, associated with higher order linear parabolic operators with variable coefficients. Here $ H(t)$ and $ J(t)$ are matrices of singular integral operators. The key to the result is an algebraic inequality involving matrices similar to the symbol of $ H(t)$ having their eigenvalues contained in a fixed compact subset of the open left-half complex plane. Then a sharp estimate on the norms of certain imbedding maps is obtained. These estimates along with the energy inequality is applied to the Cauchy problem for higher order linear parabolic operators restricted to slabs in $ {R^{n + 1}}$.

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Keywords: Partial differential operator, evolution operator, singular integral operator, symbol of an operator, test function, temperate weight function, compact support, Sobolev space
Article copyright: © Copyright 1972 American Mathematical Society