## The Asymptotics of the Laplacian on a Manifold With Boundary (1990)

Citations: | 63 - 21 self |

### BibTeX

@MISC{Branson90theasymptotics,

author = {Thomas P. Branson and Peter B. Gilkey},

title = {The Asymptotics of the Laplacian on a Manifold With Boundary},

year = {1990}

}

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### Abstract

: Let P be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions. x1 Statement of results Let M m be a compact Riemannian manifold with boundary @M: Let V be a smooth vector bundle over M equipped with a connection r V : Let E be an endomorphism of V: Define P = \Gamma(\Sigma i;j g ij r V i r V j +E) : C 1 (V ) ! C 1 (V ): Every second order elliptic operator on M with leading symbol given by the metric tensor can be put in this form. Let f 2 C 1 (M): If @M = ;; then as t ! 0 + ; T r L 2 (fe \GammatP ) ' t \Gammam=2 \Sigma n t n an (f; P ) where n = 0; 1; 2; ::: ranges over the nonnegative integers. If @M 6= ;; we must impose suitable boundary conditions. Let OE 2 C 1 (V ): Dirichlet boundary conditions are BOE = OEj @M = 0: Let OE ;N be the covariant derivative of OE with respect ...