Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von Weizsäcker Model

Lu, Jianfeng; Otto, Félix

doi:10.1002/cpa.21477

Cited by 91 publications

(123 citation statements)

References 27 publications

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“…First, let us quickly prove the nonexistence of minimizers when m is large, recovering the main result in [20].…”

Section: Existence and Nonexistence For I 0 (M)mentioning

confidence: 76%

“…However, the ionization problem, which corresponds to the nonexistence of minimizers when m is large, remains mostly open. In fact, the nonexistence when V ≡ 0 is already surprisingly delicate and has been solved recently by Lu and Otto [20]. Their proof is based crucially on the translation-invariance of E 0 (u) and does not apply to the general case.…”

Section: Introductionmentioning

confidence: 99%

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Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

Nam

Bosch

2017

Math Phys Anal Geom

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Abstract. We study the ionization problem in the Thomas-FermiDirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.

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“…First, let us quickly prove the nonexistence of minimizers when m is large, recovering the main result in [20].…”

Section: Existence and Nonexistence For I 0 (M)mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

Nam

Bosch

2017

Math Phys Anal Geom

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“…We note thatũ ε (x) ≤ u ε (x) for every x ∈ R 3 . Furthermore, for sufficiently small δ we have H ρ ≥ 0 and 45) for some universal c > 0 (where Γ is the Newtonian potential in R 3 , as above). From (5.42), (5.44) and (5.45) we then get 46) for ε small enough.…”

Section: Equidistribution Of Energymentioning

confidence: 89%

“…Importantly, the model in (1.1) is a paradigm for the energy-driven pattern forming systems in which spatial patterns (global or local energy minimizers) form as a result of the competition of short-range attractive and long-range repulsive forces. This is why this model and its generalizations attracted considerable attention of mathematicians in recent years (see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], this list is certainly not exhaustive). In particular, the volume-constrained global minimization problem for (1.1) in the whole space with no neutralizing background, which we will also refer to as the "self-energy problem", has been investigated in [34,37,45,49].…”

Section: -3mentioning

confidence: 99%

“…Here, however, we encounter a difficulty that it is not known that the minimizers of the self-energy do not exist for large enough masses. Such a result is only available in three space dimensions for the Coulombic kernel [37,45]. In the absence of such a non-existence result one may not exclude a possibility of a network-like structure in the macroscopic limit.…”

Section: -3mentioning

confidence: 99%

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Low Density Phases in a Uniformly Charged Liquid

Knüpfer

Muratov

Novaga

2016

Commun. Math. Phys.

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This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase.Under periodic boundary conditions, we prove that the considered energy Γ-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

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On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

Knuepfer

Muratov

2013

Comm Pure Appl Math

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This paper is the continuation of a previous paper (H. Knüpfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129–1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc.

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Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von Weizsäcker Model

Cited by 91 publications

References 27 publications

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

Low Density Phases in a Uniformly Charged Liquid

On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

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