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.
We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $\Gamma\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.
We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over ℤ^d.
We apply recent circle tangency estimates due to Pramanik–Yang–Zahl to prove sharp weighted Fourier extension estimates for the cone in ℝ^3 and 1-dimensional weights. The idea of using circle tangency estimates to study Fourier extension of the cone is originally due to Tom Wolff, who used it in part to prove the first decoupling estimates. We make an improvement to the best known Mizohata–Takeuchi-type estimates for the cone in ℝ^3 and the 1-dimensional weights as a corollary of our main theorem, where the previously best known bound follows as a corollary of refined decoupling estimates.
We are concerned with positive normalized solutions (u, λ) ∈ H1 (ℝ2) × ℝ to the following semi-linear Schrödinger equations -Δ u+λ u=f(u), in ℝ^2, satisfying the mass constraint ∫_ℝ^2| u|^2dx=c^2 . We are interested in the so-called mass-mixed case in which f has L2-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain-pass type. We also investigate the asymptotic behavior of solutions approaching the zero-mass case, namely when c → 0+.
The momentum formulation of the surface quasi-geostrophic equations consists of two nonlinear terms, besides the pressure term, one of which cannot be written in a divergence form. When the anti-divergence operator is applied to such nonlinear terms, in general, one cannot take advantage of the differentiation operator of order minus one unless the nonlinear terms are compactly supported away from the origin in Fourier frequency. Moreover, the two nonlinear terms of the momentum surface quasi-geostrophic equations are one derivative more singular than that of the Navier-Stokes equations. Upon employing the convex integration technique to the random partial differential equations corresponding to the momentum surface quasi-geostrophic equations forced by linear multiplicative noise, these issues create various difficulties, unseen in the deterministic scenario and even in the case the noise is additive. By making a key observation in case the solution is a shear flow and rewriting the difficult stochastic commutator error in terms of the oscillation error, we prove its non-uniqueness in law.
We show that every Hardy field extends to an ω-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.
Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping f Ω→ℝ^n with a (K, Σ)-quasiregular value at f(x_0) can be rescaled at x_0 to a non-constant K-quasiregular mapping. Our proof of this fact involves establishing a quasiregular values -version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small K -theorem is obtained as an immediate corollary of our main result.
We establish a fourth order sharp Sobolev trace inequality on three-balls, and its equivalence to a third order sharp Sobolev inequality on two-spheres.
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed measures of three convex bodies and obtained its integral representation. In this work, we obtain inequalities for mixed measures, such as a generalization of Fenchel's inequality; this provides a new, simpler proof of the classical volume case. Moreover, we show that mixed measures are connected to the study of log-submodularity and supermodularity of the measure of Minkowski sums of convex bodies. This elaborates on the recent investigations of these properties for the Lebesgue measure. We conclude by establishing that the only Radon measures that are supermodular over the class of compact, convex sets are multiples of the Lebesgue measure. Motivated by this result, we then discuss weaker forms of supermodularity by restricting the class of convex sets.
We prove a wavelet T(1) theorem for compactness of multilinear Calderón-Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator.
Hom shifts form a class of multidimensional shifts of finite type (SFT) and consist of colorings of the grid Z2 where adjacent colours must be neighbors in a fixed finite undirected simple graph G. This class includes several important statistical physics models such as the hard square model. The gluing gap measures how far any two square patterns of size n can be glued, which can be seen as a measure of the range of order, and affects the possibility to compute the entropy (or free energy per site) of a shift. This motivates a study of the possible behaviors of the gluing gap. The class of Hom shifts has the interest that mixing type properties can be formulated in terms of algebraic graph theory, which has received a lot of attention recently. Improving some former work of N. Chandgotia and B. Marcus, we prove that the gluing gap either depends linearly on n or is dominated by log(n). We also find a Hom shift with gap {\Theta}(log(n)), infirming a conjecture formulated by R. Pavlov and M. Schraudner. The physical interest of these results is to better understand the transition from short-range to long-range order (respectively sublogarithmic and linear gluing gap), which is reflected in whether some parameter, the square cover, is finite or infinite.
We construct symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on two types of polygon carpets, which are natural generalizations of planner Sierpinski carpets (SC). The first ones are called perfect polygon carpets that are natural analogs of SC in that any intersection cells are either side-to-side or point-to-point. The second ones are called bordered polygon carpets which satisfy the boundary including condition as SC but allow distinct contraction ratios.
A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen representation and reduces the complexity of the spectral problem. Yet, such a quotient operator is not uniquely defined. Here we present a computationally simple way of choosing a special basis for the space of intertwiners, allowing us to construct a quotient that reflects the structure of the original operator. This quotient construction generalizes previous definitions for discrete graphs, which either dealt with restricted group actions or only with the trivial representation. We also extend the method to quantum graphs, which simplifies previous constructions within this context, answers an open question regarding self-adjointness and offers alternative viewpoints in terms of a scattering approach. Applications to isospectrality are discussed, together with numerous examples and comparisons with previous results.
We define a relaxed version H-f(fine) of the distortion number H-f that is used to define quasiconformal mappings. Then we show that for a BV function f is an element of BV(& Ropf;(n);& Ropf;(n)), for divided by Df divided by-a.e. x is an element of & Ropf;(n) it holds that H-f*(fine)(x) < infinity if and only if dDf/d|Df|(x) has full rank.
In this paper, we study the following Lane–Emden system with non-power nonlinearity -Δ u=| v|^p-1v[ln(e+| v|)]^ϵ in Ω, -Δ v=| u|^q-1u[ln(e+| u|)]^ϵ in Ω, u=v=0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, N ≥ 3, ϵ > 0 is a small parameter, p and q lying on the critical Sobolev hyperbola 1 p + 1 + 1 q + 1 = N - 2 N . We construct multiple blowing-up solutions based on the finite dimensional Lyapunov–Schmidt reduction method as ϵ goes to zero.
