Elliptic Boundary Value Problem - Eigenvalues and Eigensolutions

Eigenvalues and Eigensolutions

Another Sobolev imbedding theorem states that the inclusion is a compact linear map. Equipped with the spectral theorem for compact linear operators, one obtains the following result.

Theorem Assume that is coercive, continuous and symmetric. The map from to is a compact linear map. It has a basis of eigenvectors and matching eigenvalues such that

1. as ,
2. ,
3. whenever and
4. for all