Liouville-type theorems for a system governed by degenerate elliptic operators of fractional orders

Evolution Equations Governed by Elliptic Operators

Liouville Type Results for Periodic and Almost Periodic Elliptic Operators

The main feature of this paper concerns extensions of the Liouville theorem to the following class of elliptic equations in non-divergence form: aij(x)∂iju + bi(x)∂iu + c(x)u = 0 in R N , with c ≤ 0. We show that the Liouville property holds (that is, the space of bounded solutions has at most dimension one) if the coefficients aij , bi and c are periodic, with the same period, and it does not ...

Analysis of degenerate elliptic operators of Grušin type

We analyze degenerate, second-order, elliptic operators H in divergence form on L2(R ×Rm). We assume the coefficients are real symmetric and a1Hδ ≥ H ≥ a2Hδ for some a1, a2 > 0 where Hδ = −∇x1 cδ1,δ′ 1(x1)∇x1 − cδ2,δ′ 2(x1)∇ 2 x2 . Here x1 ∈ R , x2 ∈ R m and cδi,δ′ i are positive measurable functions such that cδi,δ′ i(x) behaves like |x| δi as x→ 0 and |x| ′ i as x→ ∞ with δ1, δ ′ 1 ∈ [0, 1〉 a...

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.

Enclosure Theorems for Eigenvalues of Elliptic Operators

are to be considered when the coefficients a,-,-, b, and c are continuous real-valued functions with &=^0, c>0 in En. The ellipticity of L implies that the symmetric matrix (a,,) is everywhere positive definite. A "solution" u of Lu = 0 is supposed to be of class C1 and all derivatives involved in (1.1) are supposed to exist, be continuous, and satisfy Lu = 0 at every point. The eigenvalue prob...