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.
Interpolation theory in the unit ball and semi-algebraic geometry yield explicit descriptions of the boundary zeros of stable polynomials. Given a polynomial $p\in \mathbb{C}[z_1, ...,z_n]$ that is zero-free in the unit ball and vanishes on the sphere along submanifolds of dimension at most one, we describe the boundary zeros $\mathcal{Z}(p)\cap\mathbb{S}_n$ in terms of peak sets for $A^\infty(\mathbb{B}_n)$. In particular, in the setting $n=2$, we achieve a characterization by proving that every accumulation point of $\mathcal{Z}(p)\cap\mathbb{S}_2$ lies in the relative interior of an one dimensional real analytic submanifold, and that these submanifolds form a foliation of the non-isolated part of $\mathcal{Z}(p)\cap\mathbb{S}_2$. As an application of the developed theory, we obtain a characterization of cyclic polynomials without weak essential singularities in the Dirichlet-type space $\mathcal{D}_{n-1/2}(\mathbb{B}_n)$. A theory for more general geometric settings of the boundary zeros is also developed, aiming to provide a starting point for further extensions.
An $\infty$-harmonic function is a viscosity solution of $\nabla^2 u(\nabla u,\nabla u)=0$, or equivalently, an absolute minimizer of $\|\nabla u\|_{L^\infty}$. We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$-harmonic functions in domains of $\mathbb{R}^2$ are $C^{1,1/3}$. 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$-harmonic functions to inverse mean curvature flow (IMCF) clusters -- which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p\to\infty$ limit of the classical duality between $p$-harmonic and $q$-harmonic functions in $\mathbb{R}^2$, where $\frac1p+\frac1q=1$.
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Amp\`ere solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.
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.
We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable L\'evy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+k\delta_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_\mu$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.
Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.