and Applied Analysis 3 we refer to 20–22 , but the results on the boundary blow-up solutions for p x -Laplacian equations are rare see 16 . In 16 , the present author discussed the existence and asymptotic behavior of boundary blow-up solutions for the following p x -Laplacian equations: −Δp x u f x, u 0, in Ω, u x −→ ∞, as d x, ∂Ω −→ 0, 1.8 on the condition that f x, · satisfies polynomial gro...

Let Ω = ⊕∞ i=1 Z3 and ei = (0, . . . , 0, 1, 0, . . . ) where the 1 occurs in the i-th coordinate. Let F = {α ⊂ N : ∅ 6= α, α is finite}. There is a natural inclusion of F into Ω where α ∈ F is mapped to eα = ∑ i∈α ei. We give a new proof that if E ⊂ Ω with d∗(E) > 0 then there exist ω ∈ Ω and α ∈ F such that {ω, ω + eα, ω + 2eα} ⊂ E. Our proof establishes that for the ergodic reformulation of ...

The study of extremal Kähler metric is initiated by the seminal works of Calabi [4], [5]. Let (M, [ω]) be a compact Kähler manifold with fixed Kähler class [ω]. For any Kähler metrics g in the fixed Kähler class [ω], the Calabi energy C(g) is defined as C(g) = M s 2 dµ, where s is the scalar curvature of g. The extremal Kähler metric is the critical point of the Calabi energy. The Euler-Lagrang...

Two further equivalent axioms are given for valuations of a matroid. Let M = (V,B) be a matroid on a finite set V with the family of bases B. For ω : B → R the following three conditions are equivalent: (V1) ∀B,B′ ∈ B, ∀u ∈ B −B′,∃v ∈ B′ −B: ω(B) + ω(B′) ≤ ω(B − u+ v) + ω(B′ + u− v); (V2) ∀B,B′ ∈ B with B 6= B′, ∃u ∈ B −B′,∃v ∈ B′ −B: ω(B) + ω(B′) ≤ ω(B − u+ v) + ω(B′ + u− v); (V3) ∀B,B′ ∈ B, ∀...

Let Ω be a Borel subset of SN where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti’s theorem characterizes these measures when Ω = SN. We apply the ergodic theory of equivalence relations to study the case Ω 6= SN, and obtain versions of this theorem ...

We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely. 1. A model problem The g...

We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents  −div(p(x)∇u) = β|u|2−2u+ f(x, u) in Ω, u 6≡ 0 in Ω, ∂u ∂ν = Q(x)|u| 2∗−2u on ∂Ω, where β ≥ 0, Q is continuous on ∂Ω, p ∈ H(Ω) is continuous and positive in Ω̄ and f is a lower-order perturbation of |u|2−1 with f(x, 0) = 0.