In this paper, we show that as τ → √−1∞, any zero of the Lam´e function converges to either ∞ or a finite point p satisfying Rep = 1 2 and e2πip being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line Rez = 1 2. We also compute explicitly the eigenvalue setofthecorresponding L2[0,1] eigenvalueproblemforsuchKdVpotentials, thus extends Takemura’s works [26, 27]. Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy, Unterkofler and Weikard [11] and the Darboux transformation.

We improve the estimates in the restriction problem in dimension n⩾4. To do so, we establish a weak version of a k-linear restriction estimate for any k. The exponents in this weak k-linear estimate are sharp for all k and n.

We obtain improved fractional Poincaré inequalities in John domains of a metric space (X,d) endowed with a doubling measure μ under some mild regularity conditions on the measure μ. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the nonintegrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

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