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Rupert L. Frank

Publications and source records attributed to Rupert L. Frank.

At least 19 recordsLinked to original sources
arXiv2026

Large particle number asymptotics for point-interacting bosons in the plane

We consider $N$ bosons in two dimensions interacting via point interactions, modeled by the TMS Hamiltonian. We prove that their ground state energy has the asymptotic behavior $-e^{C N }$ for large $N$, for an explicit constant $C$ related to the sharp constant in the Ladyzhenskaya-Gagliardo-Nirenberg inequality. Such asymptotic behavior was predicted in the physics literature, based on a variant of the mean-field approximation.

math-ph↗
arXiv2026

A new sharp Sobolev inequality between the $Q$-curvature and the scalar curvature on Euclidean space

We establish a sharp Sobolev-type inequality on $\mathbb{R}^n$, $n\geq 5$, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant $S_{4,2}$ in the inequality \begin{equation*} \int_{\mathbb R^n} (Δu)^2\,dx \ge S_{4,2} \left(\int_{\mathbb{R}^n}|\nabla u|^{2}|u|^{\frac{4}{n-4}}\,dx\right)^{\frac{n-4}{n-2}}, \end{equation*} for any $u\in\dot H^2(\mathbb{R}^n)$, and we classify the cases of equality. This solves the problem of minimizing the total $Q$-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.

math.AP↗
arXiv2026

Optimal Riesz mean inequalities for the Coulomb Hamiltonian

We are interested in whether the Riesz means of the eigenvalues of the Coulomb Hamiltonian can be bounded by the corresponding semiclassical phase space integral. In dimensions two and three we determine the optimal Riesz exponent above which this bound holds.

math.SP↗
arXiv2026

The sharp constant in the Chang--Marshall inequality

We prove that the optimal constant in the Chang--Marshall inequality is $e$, as conjectured by Andreev and Matheson. More precisely, if $f$ is analytic in the unit disk, vanishes to order $m\geq1$ at the origin, and has normalized Dirichlet integral $d\leq1$, then the boundary mean of $e^{|f|^2}$ is at most $e^{d/m}$. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.

math.CV↗
arXiv2026

Sharp Sobolev inequalities under first moment constraints

Let $n\geq2$ and $1<p<n$. We study sharp Sobolev inequalities on the round sphere $\mathbb{S}^n$ for functions $u\in W^{1,p}(\mathbb{S}^n)$ satisfying $\int_{\mathbb{S}^n}x|u|^{np/(n-p)}\,dσ=0$. We prove that \[ \|u\|_{L^{np/(n-p)}(\mathbb{S}^n)}^p \leq 2^{-p/n}S_{n,p}^p\|\nabla u\|_{L^p(\mathbb{S}^n)}^p+B\|u\|_{L^p(\mathbb{S}^n)}^p \] holds for all such functions with some $B=B(n,p)<\infty$ if and only if $1<p\leq2$ or $2<p<n<3p-2$, where $S_{n,p}$ is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin's inequality holds with the sharp gradient coefficient. For every $2\leq p<n$, we also establish the corresponding inequality with squared norms and gradient coefficient $2^{-2/n}S_{n,p}^2$. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.

math.AP↗
arXiv2026

Another characterization of $\dot W^{1,p}(\mathbb R^d)$

We improve an earlier characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$, $1<p<\infty$, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.

math.FA↗
arXiv2026

The double sphere solution in the liquid drop model

We consider the problem of finding critical domains $Ω\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (Ω) = {\rm Per}\,(Ω) + \frac 12 \iint_{Ω\times Ω} \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|Ω|=V$. We look for smooth, embedded, compact surfaces $\partialΩ$ 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

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

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

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

The Winding Number at the Critical Hölder 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 $α$-Hölder continuous with $α>1/3$, 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 $α=1/3$, thereby resolving an open problem by Brezis.

math.CA↗
arXiv2026

The CKN inequality for spinors: symmetry and symmetry breaking

This paper is devoted to Sobolev interpolation inequalities for spinors, with weights of Caffarelli-Kohn-Nirenberg (CKN) type. In view of the corresponding results for scalar functions, a natural question is to determine whether optimal spinors have symmetry properties, or whether spinors with symmetry properties are linearly unstable, in which case we shall say that symmetry breaking occurs. What symmetry means has to be carefully defined and the overall picture turns out to be richer than in the scalar case. So far, no symmetrization technique is available in the spinorial case. We can however determine a range of the parameters for which symmetry holds using a detailed analysis based mostly on spectral methods.

math.AP↗
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

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

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

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↗