𝐿^{𝑝} theory of differential forms on manifolds

Scott, Chad

doi:10.1090/S0002-9947-1995-1297538-7

$L^ p$ theory of differential forms on manifolds
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by Chad Scott

Trans. Amer. Math. Soc. 347 (1995), 2075-2096

DOI: https://doi.org/10.1090/S0002-9947-1995-1297538-7

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Abstract:

In this paper, we establish a Hodge-type decomposition for the ${L^p}$ space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an ${L^p}$ estimate which contains, as a special case, the ${L^2}$ result referred to by Morrey as Gaffney’s inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the ${L^p}$ boundedness of Green’s operator which we use in developing the ${L^p}$ theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the ${L^p}$ norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the $A$-harmonic equation.

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