Introduction to Partial Differential Equations by David Borthwick

Author:David Borthwick
Language: eng
Format: epub, pdf
ISBN: 9783319489360
Publisher: Springer International Publishing

Boundary conditions for which (7.24) holds are called self-adjoint boundary conditions (for the Laplacian). Formally, (7.24) resembles the matrix condition (7.23), but of course there is no analog of boundary conditions in the matrix case. The proper definition of self-adjointness in functional analysis involves a more precise specification of the domain on which acts and (7.24) holds. Even without going into these details, we can still draw some meaningful conclusions from Lemma 7.11.

Lemma 7.12

Suppose is a sequence of eigenvalues of on a bounded domain , with eigenvectors in subject to a self-adjoint boundary condition. Then and, after possible rearrangement, the eigenvectors form an orthonormal sequence in .

Furthermore, for Dirichlet conditions, and for Neumann.

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