We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local H\"older continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schr\"odinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. The constant is best possible, even within the class of closed manifolds: it is attained identically by an explicit self-similar solution in flat $\Rn$, and its sharpness transfers to the compact setting through a large-torus limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$, $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows in which the second-order regularization and the first-order eikonal term are decoupled: for this family the maximum principle applies with the sharp constant, uniformly in the regularization parameter, and with no assumption on the critical set of the solution. The identity is moreover form-invariant under the classical $\alpha$-relaxation of the Li-Yau functional; as a consequence we also obtain the corresponding inequality on closed manifolds with $\operatorname{Ric} \ge -\kappa$ (again with no assumption beyond the Ricci lower bound), and, on complete noncompact manifolds with $\operatorname{Ric} \ge 0$, the sharp inequality for the approximating flows -- again with no assumption on the critical set -- under a cutoff hypothesis on the distance function (automatic in $\Rn$, and, for $p \ge 2$, under nonnegative sectional curvature) and a qualitative polynomial growth condition on the Li-Yau quantity, satisfied, with exponent zero, by the extremal profile. A sharp global Harnack inequality follows.
For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $\sigma_2$-Yamabe equation admits a noncompact family of positive $\Gamma_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $\sigma_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.
We develop a harmonic-coordinate approach to the identity component $G_\omega(\Sigma_g)$ of the symplectomorphism group of a closed oriented surface of genus $g\ge2$. Using an intrinsic decomposition of the Abel-Jacobi displacement into a global flux part and a zero-average harmonic fluctuation, we introduce the harmonic flux norm and prove its non-degeneracy on the full identity component without Floer theory. We also show the norm is continuous in the $C^0$-topology, deduce that $\mathrm{Ham}(\Sigma_g,\omega)$ is $C^0$-closed inside $G_\omega(\Sigma_g)$, and produce a locally injective harmonic-coordinate chart near the identity. Along the way we derive first-order expansions for the norm, quantitative fixed-point obstructions, and propose a finite-dimensional persistence invariant (the harmonic barcode) associated to the harmonic displacement filtration.
For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a M\"obius--Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15--16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.
We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing $\mathcal C_d$.