Maximum principle for higher order operators in general domains

We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

Harnack-type inequality for linear fractional stochastic equations

Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter {H<\frac{1}{2}}. We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.

Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

Let {\mathcal{L}} be the Laplace operator on {\mathbb{R}^{d}} , {d\geq 3} , or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation {\mathcal{L}u-\varphi(\,\cdot\,,u)=0} we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

A Harnack-Type Inequality for a Prescribed-Curvature Equation on a Domain with Boundary

In this paper the method of moving spheres is used to derive a Harnack-type inequality for positive solutions ofwhere n ≥ 4, ℝ.

Geometric character of domain and the Harnack inequality of solution of a singular parabolic equation

The geometric character of domains of solutions of a singular parabolic equation with the Neumann boundary condition are discussed in this work. We prove a gradient estimate and a Harnack type inequality and then, show that the domains of the solutions are exactly columns in the space-time. Specially, the altitudes of the columns are calculated accurately.