Functoriality for higher rho invariants of elliptic 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.

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...

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

On Functoriality of Homological Mirror Symmetry of Elliptic Curves

The mirror functor as constructed by Polishchuk and Zaslow depends on various choices. This calls for a description of how the constructed functor depends on these choices. For a “good” mirror functor Φτ we would expect that we get a symplectic counterpart, that has a geometric explanation in terms of complexified symplectic tori. But unfortunately, the mirror functor does not have this expecte...