An L2 model for selfadjoint elliptic differential operators with constant coefficients on bounded domains

The spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions

Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators

A Morse Index Theorem for Elliptic Operators on Bounded Domains

We consider a second-order, selfadjoint elliptic operator L on a smooth one-parameter family of domains {Ωt}t∈[a,b] with no assumptions on the geometry of the Ωt’s. It is shown that the Morse index of L can be equated with the Maslov index of an appropriately defined path in a symplectic Hilbert space constructed on the boundary of Ωb. Our result is valid for a wide variety of boundary conditio...

A Vanishing Conjecture on Differential Operators with Constant Coefficients

In the recent progress [BE1], [Me] and [Z2], the wellknown JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd or...

on the spectral properties of degenerate non-selfadjoint elliptic systems of differential operators