A Borg–Levinson Theorem for Higher Order Elliptic Operators

Lagrange Multipliers for Higher Order Elliptic Operators

In this paper, the Babuška’s theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions. Mathematics Subject Classification. 41A10, 41A17, 65N15, 65N30. Received: April 5, 2004.

On the Bethe-Sommerfeld conjecture for higher-order elliptic operators

We consider the elliptic operator P(D)+ V in R , d ≥ 2 where P(D) is a constant coefficient elliptic pseudo-differential operator of order 2 with a homogeneous convex symbol P(ξ), and V is a real periodic function in L∞(Rd). We show that the number of gaps in the spectrum of P(D)+V is finite if 4 > d + 1. If in addition, V is smooth and the convex hypersurface {ξ ∈ R : P(ξ) = 1} has positive Ga...

Local Hardy Spaces Associated with Inhomogeneous Higher Order Elliptic Operators

Let L be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this article, for all p ∈ (0, ∞) and L satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces hpL(R ) associated with L, which coincide with Goldberg’s local Hardy spaces h(R) for all p ∈ (0, ∞) when L ≡ −∆ (the Laplace operator). The authors al...

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

Elliptic Operators and Higher Signatures

Building on the theory of elliptic operators, we give a unified treatment of the following topics: • the problem of homotopy invariance of Novikov’s higher signatures on closed manifolds; • the problem of cut-and-paste invariance of Novikov’s higher signatures on closed manifolds; • the problem of defining higher signatures on manifolds with boundary and proving their homotopy invariance.