18 citations · 22 across the 21 of their papers we have counts for
23 papers
The sharp constant in the Chang--Marshall inequality
Rupert L. Frank, Paata Ivanisvili
We prove that the optimal constant in the Chang--Marshall inequality is , as conjectured by Andreev and Matheson. More precisely, if is analytic in the unit disk, vanishes t…
Sharp comparison between the perimeter and its fractional analogue in two dimensions
Rupert L. Frank, Paata Ivanisvili
We show that the fractional perimeter of any order is maximal, among all planar sets of fixed perimeter, for the disk.
A counterexample to the Kato conjecture for positive commutators
Rupert L. Frank, Paata Ivanisvili
We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators and by showing that the op…
Counterexamples to the Landis conjecture in dimensions three and higher
Rupert L. Frank, Paata Ivanisvili
In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c|…
The Winding Number at the Critical Hölder Exponent 1/3: Failure of Universal Fourier Summation
Rupert L. Frank, Paata Ivanisvili
The degree (or winding number) of a sufficiently regular map is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\…
Pleijel's theorem for a class of degenerate elliptic operators
Rupert L. Frank, Bernard Helffer
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…
The distance between homotopy classes of Sobolev maps on spheres
Rupert L. Frank, Paata Ivanisvili
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 eq…
Minimizers for Coulomb gases constrained to a halfspace
Rupert L. Frank, Paata Ivanisvili, Clara Torres-Latorre
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…
The sharp extension norm for a planar sector
Rupert L. Frank, Paata Ivanisvili
We compute explicitly the infimum of the norms of -extension operators for planar sectors and exhibit an extension operator attaining this infimum. This solves an open pro…
Sobolev extensions, interpolation inequalities and consequences
Rupert L. Frank, Pekka Koskela, Riddhi Mishra
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 applica…
The sharp log Sobolev inequality on finite cycles
Rupert L. Frank, Paata Ivanisvili
We settle the problem of finding the sharp constant in the log Sobolev inequality on the -cycle for all , by showing that it is equal to half of the spectral gap. We ded…
Eigenvalue asymptotics of Müller minimizers for atoms and molecules
Rupert L. Frank, Long Meng, Phan Thành Nam +1
We study the spectral properties of minimizers of the Müller functional for atoms and molecules with electrons and total nuclear charge . We prove that under some suitable a…
An asymptotic shape optimization problem for Riesz means of Laplacian eigenvalues
Rupert L. Frank, Simon Larson
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 def…
Sharp upper bounds for the density of relativistic atoms: Noninteracting case
Rupert L. Frank, Konstantin Merz
We prove an optimal upper bound for the density of electrons of an infinite Bohr atom (no electron-electron interactions) described by the relativistic operators of Chandrasekhar a…
Uniform bounds for Neumann heat kernels and their traces in convex sets
Rupert L. Frank, Simon Larson
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…
A Jensen inequality for partial traces and applications to partially semiclassical limits
Eric A. Carlen, Rupert L. Frank, Simon Larson
We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application we reprove and extend some theorems about eigenvalue asymptotics of…
Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian
Rupert L. Frank, Simon Larson, Paul Pfeiffer
The Berezin--Li--Yau and the Kröger inequalities show that Riesz means of order of the eigenvalues of the Laplacian on a domain of finite measure are bounded in terms…
Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups
Rupert L. Frank, Bernard Helffer, Ari Laptev
We show that the -th Dirichlet eigenvalue of the sub-Laplacian on an open set of a Carnot group is greater than the -st Neumann eigenvalue. This extends earlier results i…
Spectral asymptotics for Robin Laplacians on Lipschitz sets
Rupert L. Frank, Simon Larson
We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as…
Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains
Rupert L. Frank, Simon Larson
We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that th…
Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains
Rupert L. Frank, Simon Larson
We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary po…
On Courant and Pleijel theorems for sub-Riemannian Laplacians
Rupert L. Frank, Bernard Helffer
We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, wh…
Eigenvalue estimates for Schroedinger operators on metric trees
Tomas Ekholm, Rupert L. Frank, Hynek Kovarik
We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these ineq…