Das Picard-Prinzip und verwandte Fragen bei St�rung von harmonischen R�umen

Sub-Gaussian estimates of heat kernels on infinite graphs

“…However, this claim becomes rather nontrivial for arbitrary graphs (and manifolds) because of topological difficulties. We provide here a new, simple, and general proof of the implication (G) =⇒ (H ), which is based on the potential theoretic approach of A. Boukricha [12].…”

Picard principle for negative planar potentials

“…e.g. [2,3,14], etc. ): g (x; y) = g (y; x); g is lower semicontinuous on G × G and is continuous on G × G\{(x; x) : x ∈ G}; for each y ∈ G; g ( · ; y) is a -potential on G and -harmonic on G\{y}.…”

“…3 below). According to Bouligand [4] and also Brelot [5] we say that the Picard principle is valid for on a if there is a function u in H ( a ; @ a ) + such that H ( a ; @ a ) + = { u : ∈ R + } ; (2) where R is the real number ÿeld and R + ={ ∈ R : = 0}. Here it may happen that u ≡ 0 on a so that H ( a ; @ a ) + = {0}.…”

Dirichlet integral and Picard principle

“…For every t in (0,1) consider the subregion S t : e(z) < t of Ω, then S t j 0 as t -• 0. Moreover from (1) it follows that dn 0 ^ ί e{z)P(z)dxdy = f…”

“…e a0 (e aί , resp.) may be represented in terms of E a on (0, exp(-pj) as follows: (r) e aQ (exp (-p a )) " \ log r' 1 and hence in view of (8) and Theorem in no. 12 we deduce P a e^(α> 0) and P o e 2 Λ -@ % , where P o = P a with a = 0.…”