Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems

Steklov approximations of harmonic boundary value problems on planar regions

“…In particular, that analysis shows that each δ j is of finite multiplicity and δ j → ∞ as j → ∞; see Theorem 7.2 of [1]. The maximizers not only are ∂−orthonormal but they also satisfy…”

“…The proof that this norm is equivalent to the usual (1, 2)−norm on H 1 (Ω) when (B1) holds is Corollary 6.2 of [1] and also is part of theorem 21A of [19].…”

“…The existence and some properties of such eigenfunctions are described in sections 6 and 7 of [1] for a more general system. In particular, that analysis shows that each δ j is of finite multiplicity and δ j → ∞ as j → ∞; see Theorem 7.2 of [1].…”

“…These eigenvalues and a corresponding family of ∂−orthonormal eigenfunctions may be found using variational principles as described in sections 6 and 7 of Auchmuty [1]. δ 0 = 0 is the least eigenvalue of this problem corresponding to the eigenfunction s 0 (x) ≡ 1 on Ω.…”

“…These are based on the observation that there is an algorithm for constructing a basis of classes of harmonic functions on regions consisting of harmonic Steklov eigenfunctions. See Auchmuty [1]- [3] and the boundary traces of these eigenfunctions are bases of associated Hilbert spaces of functions on ∂Ω. In particular various error estimates for Steklov approximations are obtained.…”