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Rupert L. Frank

Publications and source records attributed to Rupert L. Frank.

At least 19 recordsLinked to original sources
arXiv2026

The double sphere solution in the liquid drop model

We consider the problem of finding critical domains $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.

math.AP↗
arXiv2026

A counterexample to the Kato conjecture for positive commutators

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

math.FA↗
arXiv2026

The Winding Number at the Critical H\"older Exponent 1/3: Failure of Universal Fourier Summation

The degree (or winding number) of a sufficiently regular map $f:\mathbb{T} \to \mathbb {S^1}$ is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\mathbb{Z}} n\,|\widehat{f}(n)|^2. $$ At lower regularity the series may diverge, but, as shown by Kahane, when $f$ is $\alpha$-H\"older continuous with $\alpha>1/3$, then the degree can be recovered by a universal linear summation process. We show that no summation process satisfying Brezis's natural axioms can recover the degree universally for $\alpha=1/3$, thereby resolving an open problem by Brezis.

math.CA↗
arXiv2026

Stability for the Affine Sobolev Inequality and its Critical Points for $p\ge 2$

We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We also show a corresponding result for critical points of the functional in the absence of bubbling. An important ingredient in our proof is the classification of positive energy solutions to the critical affine $p$-Laplace equation.

math.AP↗
arXiv2026

The distance between homotopy classes of Sobolev maps on spheres

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees. In the case of the 2-sphere this resolves an open problem by Brezis.

math.AP↗
arXiv2026

Minimizers for Coulomb gases constrained to a halfspace

We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

math.AP↗
arXiv2026

Sobolev extensions, interpolation inequalities and consequences

We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\infty$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.

math.FA↗
arXiv2026

The sharp extension norm for a planar sector

We compute explicitly the infimum of the norms of $W^{1,2}$-extension operators for planar sectors and exhibit an extension operator attaining this infimum. This solves an open problem posed by Maz'ya.

math.FA↗
arXiv2026

Pleijel's theorem for a class of degenerate elliptic operators

We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's bound for the Dirichlet Laplacian. The operators considered include the Baouendi-Grushin operator and operators with ellipticity degenerating on the boundary.

math.AP↗
arXiv2026

The sharp log Sobolev inequality on finite cycles

We settle the problem of finding the sharp constant in the log Sobolev inequality on the $n$-cycle for all $n\ge 4$, by showing that it is equal to half of the spectral gap. We deduce this result from an optimal cubic Sobolev inequality.

math.AP↗
arXiv2026

From Bogoliubov-de Gennes to Ginzburg-Landau: Critical Points Near $T_{\rm c}$ in the Non-Magnetic Case

We study the relation between the Bogoliubov-de Gennes (BdG) equation and the Ginzburg-Landau (GL) equation for a BCS model without external fields. While previous rigorous derivations of GL theory from BCS theory have focused on energies and minimizers, here we consider arbitrary critical points in the relevant energy regime. For temperatures close to the critical temperature, we prove that every sufficiently small solution of the BdG equation admits an asymptotic factorization into a microscopic Cooper-pair profile and a macroscopic order parameter. The latter satisfies the GL equation up to an error that vanishes in the scaling limit. We also prove the opposite direction, going from a solution of the GL equation to an approximate solution of the BdG equation. Our analysis relies on a Birman-Schwinger reformulation of the BdG equation, a Lyapunov-Schmidt type reduction, and semiclassical estimates at low regularity.

math.AP↗
arXiv2026

An asymptotic shape optimization problem for Riesz means of Laplacian eigenvalues

We review our recent results on the problem of optimizing Riesz means of Laplace eigenvalues among convex sets of given measure in the regime where the cut-off parameter in the definition of the Riesz means tends to infinity. We show that for a certain range of Riesz exponents, the optimizing sets converge to a ball. We also present some new results where we optimize over disjoint unions of convex sets.

math.SP↗
arXiv2026

Eigenvalue asymptotics of M\"uller minimizers for atoms and molecules

We study the spectral properties of minimizers of the M\"uller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a M\"uller minimizer $\gamma_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $\gamma_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schr\"odinger ground states, the analysis in M\"uller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.

math-ph↗
arXiv2026

Uniform bounds for Neumann heat kernels and their traces in convex sets

We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends on the underlying domain only through very simple geometric characteristics. This is proved via a precise and uniform expansion of the on-diagonal heat kernel close to the boundary. Most of our results are valid without the convexity assumption and we also consider two-term asymptotics for the heat trace for Lipschitz domains.

math.AP↗
arXiv2025

Jost solutions and direct scattering for the continuum Calogero-Moser equation

We propose an inverse scattering transform for the continuum Calogero-Moser equation. We give a rigorous treatment of the direct scattering problem by constructing the associated Jost solutions and introducing a distorted Fourier transform, as well as deriving trace formulas for the eigenvalues of the Lax operator.

math.AP↗
arXiv2025

Openness of shape optimizers in higher-order and non-scalar problems

We consider the first eigenvalues of the polyharmonic, Lam\'e and Stokes operators with Dirichlet boundary conditions on sets of given finite measure. It is shown that a quasi-open set for which this eigenvalue is minimal is open. This removes dimensional restrictions in earlier works. We use Campanato theory, which works well in the present higher order or non-scalar setting.

math.AP↗