In this paper, we study the characterization of inner uniformity of bounded domains G in ℝn, and prove that the following three conditions are equivalent: (1) G is inner uniform; (2) G is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; (3) G is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions (1) and (2), and the implication from (2) to (3) affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.
The paper investigates the well-posedness and the complete regularity of the weak solutions, and the longtime dynamics for the structurally damped wave equation with almost-linear h(x, ut) and supercritical nonlinearity g(x, u) on ℝN (N ⩾ 3): utt − Δu + (− Δ)αut + h(x, ut) + g(x, u) = f, where the perturbed parameter α ∈ (1/2, 1) is a dissipative index determining the dissipative strength. We show that when the growth order p of the nonlinearity g(x, u) is up to the supercritical range: p^∗
We are concerned with a critical Choquard system with prescribed mass -Δ u+λ_1u=(I_μ∗| u|^2_μ^∗)| u|^2_μ^∗-2u+ν p(I_μ∗| v|^q)| u|^p-2u in ℝ^N, -Δ v+λ_2v=(I_μ∗| v|^2_μ^∗)| v|^2_μ^∗-2v+ν q(I_μ∗| u|^q)| v|^p-2v in ℝ^N, ∫_ℝ^Nu^2=a^2, ∫_ℝ^Nv^2=b^2, where N ≥ 3, 0 < μ < N, ν ∈ ℝ, Iμ: ℝN → ℝ is a Riesz potential, 2_μ^∗:=2N-μN-2 and 2N-μN-2
We investigate the existence, asymptotic boundary behavior and uniqueness of viscosity solutions u ∈ C0(Ω) of equations M_𝐚(D^2u)=f(u)+h(x) in Ω ⊂ ℝn such that u(x) → ∞ as x → ∂Ω. Such solutions are referred to as large or boundary blow-up solutions. Here, Ω is a smooth bounded domain, M_𝐚 is a weighted partial trace operator, f is a non-decreasing function that satisfies the Keller–Osserman condition, and h is a continuous function in Ω. The main difficulty in the investigation rests on the possibility that M_𝐚 is very degenerate elliptic, and h is unbounded as well as sign-changing in Ω. To the best of our knowledge, large solutions to equations involving partial trace operators have not been investigated before.
We characterize the sequences nj of integers for which, for every finite continuous Borel measure μ on [0, 1], the Cesàro averages of the sequence {μ̂(n_j)} converge to 0 (where μ̂(n)=∫_0^1exp(-2π inx)dμ(x) stands for the n-th Fourier–Stieltjes coefficient of μ). Some relevant problems on distribution modulo 1 of real sequences are studied. The slow convergence of functions is introduced and used as a tool. The proof of the equivalence of several definitions of slow convergence utilizes H. P. Rosenthal’s combinatorial result.
