On Some Open Problems in Many-Electron Theory

Mel Levy and Elliott Lieb are two of the most prominent researchers who have dedicated their efforts to the investigation of fundamental questions in many-electron theory. Their results have not only revolutionized the theoretical approach of the field, but, directly or indirectly, allowed for a quantum jump in the computational treatment of realistic systems as well. For this reason, at the conclusion of our book where the subject is treated across different disciplines, we have asked Mel Levy and Elliott Lieb to provide us with some open problems, which they believe will be a worth challenge for the future also in the perspective of a synergy among the various disciplines.Comment: "Epilogue" chapter in "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View", Volker Bach and Luigi Delle Site Eds. pages 411-416; Book Series: Mathematical Physics Studies, Springer International Publishing Switzerland, 2014. The original title has been modified in order to clarify the subject of the chapter out of the context of the boo

The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with Cut-off Reaction Rate. I. Permanent Form Travelling Waves

We consider Kolmogorov--Petrovskii--Piscounov (KPP) type models in the presence of a discontinuous cut-off in reaction rate at concentration $u=u_c$. In Part I we examine permanent form travelling wave solutions (a companion paper, Part II, is devoted to their evolution in the large time limit). For each fixed cut-off value $0<u_c<1$, we prove the existence of a unique permanent form travelling wave with a continuous and monotone decreasing propagation speed $v^*(u_c)$. We extend previous asymptotic results in the limit of small $u_c$ and present new asymptotic results in the limit of large $u_c$ which are respectively obtained via the systematic use of matched and regular asymptotic expansions. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut-off Fisher reaction function

Isoperimetric inequalities for some integral operators arising in potential theory

In this paper we review our previous isoperimetric results for the logarithmic potential and Newton potential operators. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of operators on arbitrary domains. We demonstrate these in explicit examples.Comment: This conference paper gives a review of our previous results in the subjec

NONEXISTENCE RESULTS FOR SOLUTIONS OF SEMILINEAR ELLIPTIC-EQUATIONS

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On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were madeTo appear in Bull. Math. SciThe authors were supported in parts by the EPSRC Grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK Grant 5127/GF4

Sharp Fractional Hardy Inequalities in Half-Spaces

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.Comment: 6 pages; dedicated to V. G. Maz'y