Uniform boundary Harnack principle and generalized triangle property

Abstract

Let D be a bounded open subset in R d , d 2, and let G denote the Green function for D with respect to (− ) /2 , 0 < 2, < d. If < 2, assume that D satisfies the interior corkscrew condition; if =2, i.e., if G is the classical Green function on D, assume-more restrictively-that D is a uniform domain. Let g = G(·, y 0 ) ∧ 1 for some y 0 ∈ D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that G(z, y)/g(z) C G(x, y)/g(x) when d (z, x) d(z, y). An intermediate step is the approximation G(x, y) ≈ |x − y| −d g(x)g(y)/g(A) 2 , where A is an arbitrary point in a certain set B(x, y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D ×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.

Time-dependent gradient perturbations of fractional Laplacian

“…Every change of the order of the integration is justifies by (16). By (17) and (18), the last term in the sum equals…”

“…We also remark that the Green function of other generators, e.g., of the relativistic operator [10], may be studied by a perturbation technique [16,26,30] similar to the one we use in this paper. The developments hinge on the various versions of the so-called 3G Theorem for the (unperturbed) Green function of α/2 , which are obtained from the so-called boundary Harnack principle ( [14,17,21], see, e.g., [3,13] for the results and references in case of the Laplacian).…”

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

“…Note that we do not use 3G inequalities (cf. [1,7,10,19]), which were applied widely to the studies of stationary Schrödinger equations and nonlinear elliptic equations (cf. [4,6,11,14,17,22] and references therein).…”

Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains

“…A uniform domain is a domain satisfying only the interior conditions for an NTA domain (see [26,33]). Note that the conditions (1.2) and (1.3) can be extended to x, y ∈ Ω, and therefore the nontangential set with aperture θ and vertex at ξ ∈ ∂Ω defined by…”