On the Eikonal equation for degenerate elliptic operators

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

Global Irregularity for Mildly Degenerate Elliptic Operators

Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in C∞. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.

The Harnack inequality for a class of degenerate elliptic operators

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in N dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last N−1 variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coeffi...

Degenerate elliptic operators: capacity, flux and separation

Let S = {St}t≥0 be the semigroup generated on L2(R ) by a selfadjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R with Lipschitz continuous boundary ∂Ω. We prove that S leaves L2(Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the b...

L 1 - uniqueness of degenerate elliptic operators

Let Ω be an open subset of R with 0 ∈ Ω. Furthermore, let HΩ = − Pd i,j=1 ∂icij∂j be a second-order partial differential operator with domain C ∞ c (Ω) where the coefficients cij ∈ W 1,∞ loc (Ω) are real, cij = cji and the coefficient matrix C = (cij) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If