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 $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.
The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.
Matthew J. Colbrook, George Stepaniants· 3 citations
Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa>0$ must satisfy $\sigma_1\geq \kappa$. We disprove this conjecture for every $n\geq 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2t\Phi}g_{\mathbb{R}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and $\sigma_1(\mathbb{B}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.
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 Jacobian Conjecture is a known unsolved problem and it is the problem number 16 of the list''Mathematical Problems for the Next Century'', made by Stephen Smale, in 1998. The problem asks whether or not the Jacobian matrix of a polynomial mapping $F:\mathbb{C}^n\to\mathbb{C}^n$ at every point being invertible implies that $F$ is an automorphism. The case $n = 1$ is trivially true, while the case $n\geq 3$ has been recently proven to be false by a counter-example provided by Levent Alp\"oge, and the case $n = 2$ is still an open problem. In this paper, we show that, for all $n \geq 1$, there exists a non-empty Zariski dense open set $U$ such that, for all $F \in U$, if the Jacobian matrix of $F$ is invertible, then $F$ is an automorpshim.
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the even-dimensional case for $n\ge1000$. We complete the remaining even-dimensional cases. Starting from the structural identities in Frankel--Urschel Lemma 2.1, we derive an exact global defect budget and combine binary rounding with Gram projection. A refined ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a separate multi-column energy argument treats $n=6$. The order-two case follows from a direct calculation. The new even-dimensional proof has been formalized in Lean 4, with Frankel--Urschel Lemma 2.1 as its sole external mathematical input. Together with Cheng's odd-dimensional theorem, this proves the S-matrix conjecture in every dimension.
In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\u{\i} is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence $(a_m)$ satisfying $a_m\leq 7m^8$ and a function $f\in L^1(\mathbb T)$ for which $0$ is a Lebesgue point, $f(0)=0$, and the means $m^{-1}\sum_{k=1}^m S_{a_k}f(0)$ are unbounded.