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Heat content asymptotics for operators of Laplace type with Neumann boundary conditions
 Math. Z
, 1994
"... Abstract. Let P be an operator of Dirac type and let D = P 2 be the associated operator of Laplace type. We impose spectral boundary conditions and study the leading heat content coefficients for D. 1. introduction Let P be an operator of Dirac type on a vector bundle V over a compact Riemannian man ..."
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Abstract. Let P be an operator of Dirac type and let D = P 2 be the associated operator of Laplace type. We impose spectral boundary conditions and study the leading heat content coefficients for D. 1. introduction Let P be an operator of Dirac type on a vector bundle V over a compact Riemannian manifold M of dimension m with smooth boundary ∂M. Let D: = P 2 be the associated operator of Laplace type. The leading symbol γ of P defines a Clifford module structure on V. Choose an auxiliary connection ∇ on V so that ∇γ = 0. Adopt the Einstein convention and sum over repeated indices; indices i, j will range from 1 to m and index a local orthonormal frame {ei} for TM. Expand P = γi∇ei + ψP. We must impose suitable boundary conditions. As P need not admit local boundary conditions, we shall consider spectral boundary conditions; these were first introduced by Atiyah et. al. [1] to study the index theorem for manifolds with boundary. Near the boundary, normalize the local frame so that em is the inward unit geodesic vector field. Let indices a, b range from 1 to m − 1 and index the induced orthonormal frame for T∂M. Let A = −γmγa∇ea + ψA for some endomorphism ψA of V ∂M; A is of Dirac type on V ∂M with respect to the induced tangential Clifford module structure γT a: = −γmγa. For the sake of simplicity, we shall assume A has no purely imaginary eigenvalues.
HEAT CONTENT, HEAT TRACE, AND ISOSPECTRALITY
, 802
"... Abstract. We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products. 1. ..."
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Abstract. We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products. 1.
NEUMANN HEAT CONTENT ASYMPTOTICS WITH SINGULAR INITIAL TEMPERATURE AND SINGULAR SPECIFIC HEAT
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