(Eigenvalues of symmetric elliptic operator) Let be a bounded subset of with boundary. There exists an orthonormal basis of where each is a solution of the problem where is the elliptic operator with and the numbers are real and

We have established during the proof of the proposition ( Existence of weak solution for elliptic Dirichlet problem 2 ) that the operator is compact in . The is symmetric in . The rest follows from the propositions ( Spectrum of compact operator ) and ( Eigenvalues of compact symmetric operator ).