Let $\omega$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\omega(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\omega}$, with a wandering interval. This construction implies that Denjoy's 1932 theorem is sharp in regularity, unless additional restrictions are imposed on the rotation number. The construction in the special case $\omega(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $\omega(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon>0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.
The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\...
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 codime...
Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function, set $s(t)=\Phi(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later resta...
The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $\gamma\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$. Herscovici, Livshy...
We prove an infinitesimal circular version of Morera's theorem. Let $D\subset\mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^+$, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circula...
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradie...
Carlo Alberto Antonini, F. Pacella, Camilla Chiara Polvara et al.· 0 citations
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