Degenerate Anisotropic Differential Operators and Applications

Degenerate Anisotropic Differential Operators and Applications

The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banach-valued Lp spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. In the last section, some ...

Pohozaev Identities for Anisotropic Integro-differential Operators

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order 2s, with s ∈ (0, 1). These identities involve local boundary terms, in which the quantity u/d|∂Ω plays the role that ∂u/∂ν plays in the second order case. Here, u is any solution to Lu = f(x, u) in Ω, with u = 0 in R \ Ω, and d is the distance to ∂Ω.

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

Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators

We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators Lpu := −∇ ∗ L(|∇Lu| ∇Lu). If φ is a positive weight such that −Lpφ ≥ 0, then the Hardy type inequality c ∫ Ω |u| φp |∇Lφ| p dξ ≤ ∫

On Roots of Eigenpolynomials for Degenerate Exactly-Solvable Differential Operators

In this paper we partially settle our conjecture from [1] on the roots of eigenpolynomials for degenerate exactly-solvable operators. Namely, for any such operator we establish a lower bound (which supports our conjecture) for the largest modulus of all roots of its unique and monic eigenpolynomial pn as the degree n tends to infinity. The main theorem below thus extends earlier results obtaine...