Eigenvalue problem of degenerate elliptic operators

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here al...

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

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

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...

Invariant Measures Associated to Degenerate Elliptic Operators

This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of Rn. For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the c...