Orbital-free density functional theory of out-of-plane charge screening in graphene
OOrbital-free density functional theory ofout-of-plane charge screening in graphene
Jianfeng Lu , Vitaly Moroz , Cyrill B. Muratov Departments of Mathematics, Physics, and Chemistry, Duke University, Box 90320, Durham,NC 27708, USA.E-mail: [email protected] Swansea University, Department of Mathematics, Singleton Park, Swansea SA2 8PP, Wales,United Kingdom.E-mail: [email protected] Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights,Newark, NJ 07102, USA.E-mail: [email protected] 11, 2018
Abstract:
We propose a density functional theory of Thomas-Fermi-Dirac-vonWeizs¨acker type to describe the response of a single layer of graphene resting ona dielectric substrate to a point charge or a collection of charges some distanceaway from the layer. We formulate a variational setting in which the proposedenergy functional admits minimizers, both in the case of free graphene layers andunder back-gating. We further provide conditions under which those minimizersare unique and correspond to configurations consisting of inhomogeneous densityprofiles of charge carrier of only one type. The associated Euler-Lagrange equa-tion for the charge density is also obtained, and uniqueness, regularity and decayof the minimizers are proved under general conditions. In addition, a bifurcationfrom zero to non-zero response at a finite threshold value of the external chargeis proved.
1. Introduction
Graphene is a two-dimensional monolayer of carbon atoms arranged into aperfect honeycomb lattice [43]. It has received a huge amount of attention inrecent years, both as a very promising material for nanotechnology applica-tions and as a model system with pronounced quantum mechanical properties(for reviews, see [1, 13, 26]). The current interest in graphene stems from itsvery unusual electronic properties closely related to the symmetry and the two-dimensional character of the underlying crystalline lattice, into which the carbonatoms arrange themselves. A free-standing graphene layer acts as a semi-metal,in which the low energy charge carrying quasiparticles (electrons and holes) be-have to a first approximation as massless fermions obeying a two-dimensional a r X i v : . [ m a t h - ph ] J un Jianfeng Lu, Vitaly Moroz, Cyrill B. Muratov relativistic Dirac equation [20, 21, 55]. Hence, their kinetic energy is proportionalto their quasi-momentum: ε k = ± (cid:126) v F | k | , (1.1)where v F (cid:39) × cm/s is the Fermi velocity, k is the wave vector and “ ± ”stands for electrons and holes, respectively. This equation is valid for | k | (cid:28) a − ,where a (cid:39) .
42 ˚ A is the nearest-neighbor distance between the carbon atomsin the graphene lattice (without taking into account the effect of the velocityrenormalization [27, 32, 41, 45, 52, 58]).In contrast to the fermions with non-zero effective mass in the usual metals orsemiconductors, in graphene the effect of interparticle Coulomb repulsion doesnot decrease with increasing carrier density [32]. This can already be seen fromsimple dimensional considerations: according to (1.1) a single particle whosewave function is localized into a wave packet of radius ∼ r would have kineticenergy E kin ∼ (cid:126) v F /r , while the energy of Coulomb repulsion per particle (inCGS units) is E Coulomb ∼ e / ( (cid:15) d r ), where e > (cid:15) d is the effective dielectric constant in the presence of a substrate. Thus theirratio α = e / ( (cid:15) d (cid:126) v F ), which characterizes the relative strength of the Coulombicinteraction, is a constant independent of r , and, furthermore, we have α (cid:39) . (cid:15) d = 1, indicating the non-perturbative role of the Coulombic interaction inthe absence of a strong dielectric background.The scaling argument above can also be applied to an electron obeying (1.1)in an attractive potential of a positively charged ion. When the valence Z of theion increases, the potential energy E pot ∼ − Ze /r of the attractive interactionbetween the electron and the ion always overcomes the kinetic energy. At thesingle particle level this effect results in non-existence of single particle groundstates for the relativistic Dirac-Kepler problem [50], which is somewhat similarto the phenomenon of relativistic atomic collapse [39]. In a more realistic mul-tiparticle setting the situation is more complicated due to strongly correlatedmany-body effects involving both the electrons and holes. In fact, exactly howthe carriers in graphene screen a charged impurity is a subject of an ongoingdebate, with qualitatively different predictions for the behavior of the screen-ing charge density and the total electrostatic potential coming from differenttheories.Early studies of screening of the electric field from point charges in graphenego back to the work of DiVincenzo and Mele, who used a self-consistent Hartree-type model to analyze the electron response to interlayer charges in intercalatedgraphite compounds [16]. They found a surprising result that the screening elec-tron density decays as 1 /r (to within an undetermined logarithmic factor),indicating that the screening charge is considerably spread out laterally withinthe graphene layer. They also made a similar conclusion from the analysis ofthe Thomas-Fermi equations for massless relativistic fermions and contrastedit with the 1 /r behavior expected from the image charge on an equipotentialplane in the case of perfect screening. In sharp contrast, Shung performed an rbital-free density functional theory of out-of-plane charge screening in graphene 3 analysis of the dielectric susceptibility of intercalated graphite compounds us-ing linear response theory [49]. His calculation implies that in the absence ofdoping only partial screening of an impurity should occur and that the electronsystem should behave effectively as a dielectric medium due to the excitationof virtual electron-hole pairs, which has an effect of renormalizing the value of (cid:15) d (see also [3, 27, 29, 32] for further discussions). He also commented that thenonlinear effects are of major importance in the screening, which explains thedifferent results he had for linear response comparing with the Thomas-Fermiresult in [16].More recently, Katsnelson computed the asymptotic behavior of the screeningcharge density for a charged impurity within the Thomas-Fermi theory of mass-less relativistic fermions with a lattice cutoff at short scales [31]. He found thatthe screening charge density should behave as 1 / ( r ln r ) far from the impurity,refining earlier results of [16] and demonstrating the importance of nonlinearscreening effects in graphene. Fogler, Novikov and Shklovskii further consideredthe effect of an out-of-plane hypercritical charge Z (cid:29) /r behavior of thescreening charge density and constant electrostatic potential in the layer) [23].They also argued for a crossover between perfect screening in the near field tail,Thomas-Fermi screening (1 / ( r ln r ) behavior of the screening charge density and1 / ( r ln r ) decay of the electrostatic potential in the layer) in the far field tail, andpartial screening (dielectric response with no screening charge and 1 /r decay ofthe electrostatic potential) in the very far tail for certain ranges of Z and α . Wealso note that a recent result indicates that in the Hartree-Fock approximationthe relative dielectric constant of graphene is, somewhat surprisingly, equal tounity in the Hartree-Fock theory, implying that the total induced charge froma charged impurity in graphene is zero (no partial screening or effectively veryweak screening due to the slow decay) [28].The differing conclusions of the above works indicate a very delicate nature ofthe problem of screening in graphene (see also the discussion in [32] and furtherreferences therein). One reason is the precise tuning of the kinetic energy, theCoulombic attraction of electrons to the impurity and the Coulombic repulsionbetween electrons, which is already evident from the scaling argument presentedearlier. Another reason is that the studies mentioned above do not account forthe correlation effects. While it is believed that exchange does not play a sig-nificant effect in graphene, correlations between electrons and holes due to theirCoulombic attraction (excitonic effects) may have an effect on the nature of theresponse beyond random phase approximation [1, 2, 32, 41, 45, 52, 57, 58]. Finally,the third reason is that in view of the crucial role played by nonlinear and non-local effects for charge carrier behavior in graphene the analysis of the problem,both mathematical and numerical, becomes rather non-trivial.Our approach to the problem of screening of point charges by a graphenelayer is via introducing a Thomas-Fermi-Dirac-von Weizs¨acker (TFDW) typeenergy for massless relativistic fermions and studying the associated variationalproblem. The considered energy functional is a variant of an orbital-free den- Jianfeng Lu, Vitaly Moroz, Cyrill B. Muratov sity functional theory (for a recent Kohn-Sham-type density functional theorysee [44]) that models the exchange and correlation effects by renormalizing thecorresponding coefficients of the Thomas-Fermi theory for the system of non-interacting massless relativistic fermions and introducing a non-local analog ofthe von Weizs¨acker term in the usual TFDW model of a non-relativistic electrongas [34, 35]. For simplicity, we begin by treating the problem of the influence of asingle point charge + Ze located at distance d (cid:29) a away from the graphene layeron the electrons in the layer. It may either correspond to the effect of a chargeplaced on a gate separated from the graphene layer by a layer of insulator in thecontext of graphene-based nanodevices, or it may correspond to an imbeddedcharged impurity or a cluster of impurities within the dielectric substrate. Aftera suitable rescaling, the TFDW energy for graphene at the neutrality point inthe presence of an impurity takes the following form: E ( ρ ) = a (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12)(cid:12) d x + 23 (cid:90) R | ρ ( x ) | / d x − (cid:90) R ρ ( x ) (cid:0) | x | (cid:1) / d x + b (cid:90) (cid:90) R × R ρ ( x ) ρ ( y ) | x − y | d x d y. (1.2)Here ρ ( x ) is the signed particle density, with ρ > ρ < a ≥ b ≥ a = 0 we recoverthe usual Thomas-Fermi model for graphene. The case of b = 0 would corre-spond to a model system of non-interacting massless relativistic fermions in anexternal potential. The meaning of each term in (1.2) and the relation to theoriginal physical parameters is explained in Sec. 2. Let us point out the unusualnon-local nature of both the first and the last terms in (1.2). The first term in-volves the homogeneous H / ( R ) norm squared (cid:82) R |∇ u | d x of u = ρ/ | ρ | / ,while the last term involves the homogeneous H − / ( R ) norm squared of ρ .This is in contrast to the conventional TFDW models of massive non-relativisticfermions, in which the first term involves the homogeneous H norm and thelast term involves the homogeneous H − norm, respectively. The difference inthe first term has to do with the relativistic character of the dispersion relationfor quasiparticles in graphene at low energies given by (1.1), while the differencein the last term reflects the three-dimensional character of Coulomb interactionand the two-dimensional character of the charge density. We point out that avon Weizs¨acker-type term similar to the first term in (1.2) appeared in the stud-ies of stability of relativistic matter (see, e.g., [37] and references therein). Wealso note that our model is different from the ultrarelativistic Thomas-Fermi-von Weizs¨acker model studied in [7, 18, 19], where a local gradient term in thekinetic energy for massless relativistic fermions in three space dimensions wasused. An analogous term for graphene would have been (cid:82) R (cid:12)(cid:12) ∇ | ρ | / (cid:12)(cid:12) d x (seeSec. 2 for the explanation of our choice of the non-local term).The model above is easily generalized to include a collection of point chargesor a localized distribution of charges some distance away from the graphene rbital-free density functional theory of out-of-plane charge screening in graphene 5 layer. If V ( x ) = − (cid:90) R d µ ( y, z )( | z | + | x − y | ) / , (1.3)where µ ( y, z ) is a finite signed Radon measure with compact support locatedat z ≥ R , e.g., µ ( y, z ) = (cid:80) Ni =1 c i δ ( y − y i ) δ ( z − z i ) with c i ∈ R , y i ∈ R and z i ≥ i = 1 , . . . , N ( c i > E ( ρ ) = a (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12)(cid:12) d x + 23 (cid:90) R ( | ρ ( x ) | / − | ¯ ρ | / ) d x − | ¯ ρ | / sgn(¯ ρ ) (cid:90) R ( ρ ( x ) − ¯ ρ ) d x + (cid:90) R V ( x )( ρ ( x ) − ¯ ρ ) d x + b (cid:90) (cid:90) R × R ( ρ ( x ) − ¯ ρ )( ρ ( y ) − ¯ ρ ) | x − y | d x d y. (1.4)Here we also included the possibility of a net background charge density ¯ ρ ∈ R ,which can be easily achieved in graphene via back-gating, and subtracted thedivergent contributions of the background charge density to the energy.In this paper we establish basic existence results for minimizers of the energy,which is a slightly generalized version of the one in (1.4), under some generalassumptions on the potential V , which include, in particular, potentials of theform given by (1.3). We begin by developing a variational framework for theproblem and proving a general existence result among admissible ρ which maypossibly change sign, see Theorem 3.1. We also establish basic regularity anduniform decay properties of these minimizers, as well as the Euler-Lagrangeequation solved by the minimizing profile.We shall emphasize that sign-changing profiles with finite energy include, inparticular, the profiles for which the Coulomb energy term does not admit anintegral representation and shall be understood in the distributional sense, evenif the profile is a continuous function (see Example 4.1). Mathematically, thismakes the analysis of the problem particularly challenging. It is an interestingopen question whether it is possible for a sign–changing minimizer to have aCoulomb energy which does not have an integral representation.We then turn our attention to minimizers among non-negative ρ . Here weprove in Theorem 3.2 the existence of a unique minimizer in the consideredclass in the case of strictly positive background charge density ¯ ρ . Importantly,using a version of a strong maximum principle for the fractional Laplacian, wealso show that these minimizers are strictly positive and, as a consequence, alsosatisfy the associated Euler-Lagrange equation. In the next theorem, Theorem3.3, we give a sufficient condition that guarantees that the global minimizeramong all admissible ρ , including those that change sign, is given by the uniquepositive minimizer obtained in the preceding theorem.The remaining two theorems are devoted to the case of zero backgroundcharge density. In Theorem 3.4 we give an existence result for non-negative Jianfeng Lu, Vitaly Moroz, Cyrill B. Muratov minimizers, alongside with strict positivity and uniqueness. In Theorem 3.5,using a suitable version of fractional Hardy inequality, we establish a bifurcationresult for a particular problem in which the background potential is given by theelectrostatic potential of a point charge some distance away from the graphenelayer. We also illustrate the conclusion of Theorem 3.5 with a numerical example.Our paper is organized as follows. In Sec. 2, we discuss the derivation andjustification of different terms in the energy and connect our model with thephysics literature. In Sec. 3, we state our main results. In Sec. 4, we introducevarious notations and auxiliary lemmas that are used throughout the paper. InSec. 5, we formulate the precise variational setting for the minimization problem.Finally, in Sec. 6 we prove Theorems 3.1 and 3.3, and in Sec. 7 we prove Theorems3.2, 3.4 and 3.5.
2. Model
Our starting point is the following (dimensional) energy for the graphenelayer in the presence of a single positively charged impurity: E ( ρ ) = C W (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12)(cid:12) d x + 23 C TFD (cid:90) R | ρ ( x ) | / d x − Ze (cid:15) d (cid:90) R ρ ( x )( d + | x | ) / d x + e (cid:15) d (cid:90) (cid:90) R × R ρ ( x ) ρ ( y ) | x − y | d x d y, (2.1)which is a functional defined on a signed particle density ρ ( x ) in a flat graphenelayer of infinite extent, with the convention that ρ > ρ < ρ ∈ C ∞ c ( R )). The terms in (2.1) are, in order:the von Weizs¨acker-type term that penalizes spatial variations of ρ , the Thomas-Fermi-Dirac term containing both the contribution from the kinetic energy of theparticles and the Dirac-type contribution from exchange and correlations, theinteraction term between the particles and the external out-of-plane point charge+ Ze , and the Coulomb self-energy in the presence of a substrate providing aneffective dielectric constant (cid:15) d .The energy functional in (2.1) should be viewed as a semi-empirical modelin which the constants C W , C TFD and (cid:15) d are to be fitted to the experimentaldata for a particular setup. It is easy to see that for an ideal uniform gas of non-interacting massless relativistic fermions the kinetic energy contribution per unitarea is given by C | ρ | / , where C = (cid:126) v F √ π and the 4-fold quasiparticledegeneracy was taken into account (see for example [8, 14, 31, 59]). We note,however, that in real graphene the Coulombic interaction noticeably renormalizesthe Fermi velocity [27,32,41,45,52,58]. In practice the value of C based on theexperimentally observed value of v F (at the experimental length scale) includes Note that in [8, 14] and some other papers in the physics literature, a factor of sgn( ρ ) wasmistakenly added to the integrand of the Thomas-Fermi term. The resulting energy functionalis then not bounded from below and is inconsistent with the Thomas-Fermi equation.rbital-free density functional theory of out-of-plane charge screening in graphene 7 the many-body effects due to Coulombic interparticle forces. Similarly, for α (cid:28) C | ρ | / , where C = ( c α − c α ) C and both c and c weakly (logarithmically) depend onthe ratio of the experimental length scale to a [5, 52]. Therefore, in the localapproximation the combined contribution of the kinetic energy and the exchangeterm would have, to the leading order in α , the form of the second term in (2.1)with some constant C >
0. This conclusion is also confirmed by recentexperimental measurements of inverse quantum compressibility in graphene [41,58]. Using the renormalized rather than bare Fermi velocity may then eliminatethe need to consider the additional exchange and correlation terms, at leaston the local level. We also note that in contrast to the usual TFDW models ofmassive non-relativistic fermions [34, 35], in graphene the local approximation tothe exchange energy does not produce a non-convex contribution to the energy.We now explain the origin of the first term in (2.1). Recall that in the usualTFDW model of massive non-relativistic fermions the analogous von Weizs¨ackerterm takes the form C W (cid:82) (cid:12)(cid:12) ∇√ ρ (cid:12)(cid:12) d x , with the constant C W ∼ (cid:126) /m ∗ , where m ∗ is the effective mass (recall that for a single parabolic band one has ρ ≥ ρ , favoring spatially homogeneous ground state density forthe system of non-interacting particles (see also the discussion in [37]). Thechoice of the specific form of the integrand is determined by the following threerequirements:1) The energy must scale linearly with ρ .2) The energy must be the square of a homogeneous Sobolev norm of ρg ( | ρ | ),for some positive scale-free function g .3) The energy must scale as the Thomas-Fermi term under rescalings of x and ρ that preserve the total number of particles.The first requirement above reflects the extensive nature of the contributions ofindividual particles. The second requirement reflects the nature of the penaltyas a scale-free quadratic form in the Fourier space. The third requirement isto make the penalty term consistent with the local kinetic energy contributioncoming from the Thomas-Fermi term.It is clear that the von Weizs¨acker term in the usual TFDW model is theunique term consistent with all the relations above. Similarly, it is then easy tosee that in the case of massless relativistic fermions the unique choice of the vonWeizs¨acker-type term for graphene is given by the first term in (2.1). Indeed,the first two requirements above are obviously satisfied, and to check the thirdone, we see that (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | κρ ( λx ) | sgn( κρ ( λx )) (cid:1)(cid:12)(cid:12)(cid:12) d x = κλ − (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12)(cid:12) d x, (2.2) Jianfeng Lu, Vitaly Moroz, Cyrill B. Muratov and (cid:90) R | κρ ( λx ) | / d x = κ / λ − (cid:90) R | ρ ( x ) | / d x, (2.3)for any κ > λ >
0. Choosing κλ − = 1 to ensure that (cid:82) R | κρ ( λx ) | d x = (cid:82) R | ρ ( x ) | d x , we have that the right-hand sides of both (2.2) and (2.3) arerescaled by the same factor. From the dimensional considerations we expect tohave C W ∼ (cid:126) v F .Let us also discuss the presence of sgn( ρ ) in the definition of the von Weiz-s¨acker-type term in (2.1). As will be seen below, it imparts the energy withsome extra degree of symmetry and makes the energy functional in (2.1) betterbehaved mathematically, thus making it a natural modeling choice. Note thatthis issue is absent in the conventional TFDW model, since in the case of massivenon-relativistic fermions ρ corresponds to the density of a single type of chargecarriers and is, therefore, non-negative. In any case, when ρ ≥
0, i.e., when theholes are absent from the consideration, our von Weizs¨acker-type term coincideswith one that has appeared in many studies of relativistic matter and can befurther used to bound at least part of the kinetic energy of electrons from below[37].Another way to understand the origin of the von Weizs¨acker-type term in theenergy is to consider the leading order “gradient” correction to the energy of auniform system of non-interacting particles. If T ( ρ ) = C W (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12)(cid:12) d x + 23 C TFD (cid:90) R | ρ ( x ) | / d x, (2.4)is the “kinetic” part of the energy (recall, however, our discussion of the exchangeand correlation effects above), then the excess contribution of the kinetic energyto the leading order in δρ ( x ) = ρ ( x ) − ρ (i.e., the second variation δ T of T around ρ ), where ρ (cid:54) = 0 is the uniform background density, is δ T = 14 C W | ρ | − (cid:90) R |∇ δρ ( x ) | d x + 14 C TFD | ρ | − / (cid:90) R | δρ ( x ) | d x, (2.5)or, in terms of the Fourier transform δ (cid:98) ρ k of δρ ( x ) is given by δ T = 12 (cid:90) R d k (2 π ) Π − k | δ (cid:98) ρ k | , Π k = 2 C TFD | ρ | − / + C W | ρ | − | k | . (2.6)Here Π k = 2 | ρ | / C − (1 − C W C − | ρ | − / | k | ) + O ( | k | ) is the polarizabil-ity operator for our model. In the absence of interactions this operator shouldcoincide to the leading order for | k | → Π k in (2.6). However, a peculiar rbital-free density functional theory of out-of-plane charge screening in graphene 9 feature of graphene is that when both the intraband (perturbations of the Fermisurface) and the interband (formation of virtual electron-hole pairs) excitationsare considered, the intraband and the interband contributions cancel each otherout, making the total polarizability Π k of the noninteracting massless relativisticfermions independent of k for an interval of | k | around zero [32]: Π k = 2 | ρ | / √ π (cid:126) v F , | k | ≤ (cid:112) π | ρ | . (2.7)This behavior is due to the cancellation of the contribution from the two bandsof the Dirac cone because of symmetry, as discussed in [29]. It is, however, argued(for example in [3,56]) that the electron-electron interaction might lead to break-ing this symmetry and changing the asymptotic behavior so that Π k decreaseslinearly near | k | = 0. Clearly, correlation effects associated with Coulombic at-traction between electrons and holes should result in a decreased contribution tothe polarizability from the interband excitations. This would be consistent withthe TFDW model we are proposing here. Thus we are thinking of the first termin (2.1) as a non-local contribution of exchange and correlations to an orbital-freedensity functional theory beyond the usual local density approximation. In anycase, the model considered here might be viewed as a natural generic densityfunctional theory model for graphene or two-dimensional massless relativisticfermions in general.We finally discuss the rescaling of (2.1) leading to (1.2). Introduce (cid:101) x = λx, (cid:101) ρ ( (cid:101) x ) = κρ ( x ) , (cid:101) E ( (cid:101) ρ ) = γE ( ρ ) . (2.8)Then the energy functional in (2.1) becomes1 γ (cid:101) E ( (cid:101) ρ ) = C W κλ (cid:90) R (cid:12)(cid:12)(cid:12) ∇ (cid:0)(cid:112) | (cid:101) ρ (˜ x ) | sgn( (cid:101) ρ (˜ x )) (cid:1)(cid:12)(cid:12)(cid:12) d (cid:101) x + 23 C TFD κ / λ (cid:90) R | (cid:101) ρ ( (cid:101) x ) | / d (cid:101) x − Ze (cid:15) d κλ (cid:90) R (cid:101) ρ ( (cid:101) x )( λ d + | (cid:101) x | ) / d (cid:101) x + e (cid:15) d κ λ (cid:90) (cid:90) R × R (cid:101) ρ ( (cid:101) x ) (cid:101) ρ ( (cid:101) y ) | (cid:101) x − (cid:101) y | d (cid:101) x d (cid:101) y. (2.9)Taking λ = 1 /d , κ = ( (cid:15) d C TFD d/e Z ) and γ = (cid:15) d C d/ ( e Z ) , we arrive at(1.2) (after dropping tildes) with a = γC W κλ = (cid:15) d C W Ze , (2.10) b = γe (cid:15) d κ λ = Ze (cid:15) d C . (2.11)Our choice of the rescaling is dictated by the fact that d is the only length scalefor the considered problem, which can be seen from the fact that the parameters a and b of the rescaled energy are completely independent of d . Also, the unitsof ρ and E are now κ − and γ − , respectively.
3. Statement of results
We start with the energy functional (1.4) for a general background potential V ( x ), with parameters a > b >
0, and background charge ¯ ρ ∈ R . Notethat since the energy is invariant with respect to the transformation ρ → − ρ, ¯ ρ → − ¯ ρ, V → − V, (3.1)it is sufficient to consider only the case ¯ ρ ≥ V ( x ) is not a priori clear,since the term involving V ( x ) in (1.4) may not be bounded from below in thenatural function classes in which the other terms in the energy are well-defined.Nevertheless, if V ( x ) is of the form of (1.3), then it is easy to see that V ∈ ˚ H / ( R ) and, hence, the term involving V in the energy can be controlled bythe Coulomb repulsion term. Indeed, by an explicit computation we have( − ∆ ) / V ( x ) = − (cid:90) R | z | d µ ( y, z )( | z | + | x − y | ) / , (3.2)implying that ( − ∆ ) / V ( x ) is smooth and decays no slower than | x | − for theconsidered class of measures µ . Therefore, in view of the fact that V ( x ) is smoothand decays no slower than | x | − , we obtain that (cid:107) V (cid:107) H / ( R ) = (cid:90) R V ( − ∆ ) / V d x < ∞ . (3.3)In fact, our existence results below only rely on the fact that the estimate in (3.3)holds. Therefore, throughout the rest of the paper we generalize the energy in(1.4) to potentials V ∈ ˚ H / ( R ). We note that by fractional Sobolev embedding[36, Theorem 8.4], [15, Theorem 6.5], these are functions in L ( R ), so the energy E ( ρ ) in (1.4) is well-defined at least for ρ − ¯ ρ ∈ C ∞ c ( R ).Caution, however, is necessary in order to assign the meaning to the energy in(1.4) for sufficiently large admissible classes when searching for minimizers, sincethe problem is formulated on an unbounded domain and ρ − ¯ ρ does not have a signa priori. Indeed, even if the natural classes of functions to consider would consistof ρ ∈ L loc ( R ), it is not a priori clear if ρ − ¯ ρ can be interpreted as a chargedensity in the sense of potential theory (i.e., whether dµ = ( ρ − ¯ ρ ) d x can beassociated to a signed measure µ on R , making the last term in (1.4) meaningful,see Example 4.1). The latter depends on the delicate decay properties of theminimizers and will be the subject of a separate work [40]. Here we avoid thesedifficulties by introducing the induced electrostatic potential U which solvesdistributionally ( − ∆ ) / U = ρ − ¯ ρ. (3.4) rbital-free density functional theory of out-of-plane charge screening in graphene 11 We then introduce E ( ρ ) := a (cid:13)(cid:13)(cid:13) sgn( ρ ) (cid:112) | ρ | − sgn(¯ ρ ) (cid:112) | ¯ ρ | (cid:13)(cid:13)(cid:13) H / ( R ) + (cid:90) R (cid:18) | ρ ( x ) | / − | ¯ ρ | / − | ¯ ρ | / sgn(¯ ρ )( ρ ( x ) − ¯ ρ ) (cid:19) d x + (cid:104) V, U (cid:105) ˚ H / ( R ) + b (cid:107) U (cid:107) H / ( R ) . (3.5)Here (cid:104)· , ·(cid:105) ˚ H / ( R ) and (cid:107)·(cid:107) ˚ H / ( R ) are the inner product and the norm associatedwith the Hilbert space ˚ H / ( R ), respectively (for details about the functionspaces see Sec. 4.1). It is then easy to see that the definition of E ( ρ ) in (3.5)agrees with that in (1.4) when ρ − ¯ ρ ∈ C ∞ c ( R ). Note that the second line in(3.5) is always non-negative and becomes zero only for ρ = ¯ ρ .We now define the following class of functions for which the energy E definedin (3.5) is meaningful: A ¯ ρ := (cid:40) ρ − ¯ ρ ∈ ˚ H − / ( R ) : sgn( ρ ) (cid:112) | ρ | − sgn(¯ ρ ) (cid:112) | ¯ ρ | ∈ ˚ H / ( R ) (cid:41) , (3.6)in the sense that E : A ¯ ρ → R ∪ { + ∞} . To see that this class consists of functionsand not merely of distributions, define u ∈ ˚ H / ( R ) for a given ρ ∈ A ¯ ρ as u := sgn( ρ ) (cid:112) | ρ | − sgn(¯ ρ ) (cid:112) | ¯ ρ | . (3.7)Then by fractional Sobolev embedding [36, Theorem 8.4], [15, Theorem 6.5], wehave u ∈ L ( R ) and, hence, ρ ∈ L loc ( R ). In particular, the integral in thesecond line in (3.5) is locally well-defined.We begin with a general result on existence of minimizers for E in (3.5) over A ¯ ρ . Theorem 3.1.
Let ¯ ρ ∈ R , let E be defined by (3.5) with V ∈ ˚ H / ( R ) , and let inf ρ ∈A ¯ ρ E ( ρ ) < . Then there exists ρ ∈ A ¯ ρ such that E ( ρ ) = inf ρ ∈A ¯ ρ E ( ρ ) .Furthermore, ρ (cid:54)≡ ¯ ρ , ρ ∈ C / ( R ) ∩ L ∞ ( R ) and ρ ( x ) → ¯ ρ as | x | → ∞ . We note that the assumption inf ρ ∈A ¯ ρ E ( ρ ) < non-trivial minimizer. Otherwise by inspection ρ = ¯ ρ is automati-cally a minimizer. Thus, existence of minimizers for E over A ¯ ρ is guaranteed forevery V ∈ ˚ H / ( R ). Also, as a consequence of its minimizing property, the func-tion ρ ( x ) in Theorem 3.1 solves distributionally the Euler-Lagrange equationassociated with E in (3.5):0 = a ( − ∆ ) / (cid:16) sgn ρ (cid:112) | ρ | (cid:17) + (cid:112) | ρ | (cid:16) sgn ρ (cid:112) | ρ | − sgn ¯ ρ (cid:112) | ¯ ρ | + V + bU (cid:17) . (3.8)In fact, it is more natural to write (3.8) in terms of the variable u defined in(3.7) (see Sec. 6.2). Let us also mention that while H¨older regularity holds forgeneral potentials V from ˚ H / ( R ), if ρ changes sign one may not be able to obtain arbitrarily high regularity of ρ for smooth potentials V like in (1.2), seeRemark 6.2.While the result in Theorem 3.1 gives a very general existence result, it pro-vides only a few basic properties of the minimizers. In particular, it is not apriori clear whether ρ has a sign, even for the potential due to a single chargedimpurity appearing in the definition of E in (1.2). This is not merely a technicalissue, since in graphene one generally needs to account for the presence of bothelectrons and holes, especially at the neutrality point, i.e., when ¯ ρ = 0. It wouldseem plausible, however, that in certain situations the minimizers consist onlyof the charge carriers of one type. We speculate that this may indeed be the casefor the minimizers of E in (1.2) for all values of the parameters. At least inthe asymptotic limits a → b → ∞ the minimizers of E are expected to bepositive. We caution the reader, however, that in general the situation is ratherdelicate, since, even for a negative V with nice decay properties at infinity, theminimizer might still change sign [40].Motivated by the above observations, for ¯ ρ ≥ ρ ≥
0, which implies that there are only electrons inthe graphene layer: A +¯ ρ := (cid:110) ρ − ¯ ρ ∈ ˚ H − / ( R ) : √ ρ − √ ¯ ρ ∈ ˚ H / ( R ) , ρ ≥ (cid:111) . (3.9)Within this admissible class, we have the following counterpart of Theorem 3.1in the case of strictly positive background charge. Theorem 3.2.
Let ¯ ρ > , let E be defined by (3.5) with V ∈ ˚ H / ( R ) and let V (cid:54)≡ . Then there exists a unique ρ ∈ A +¯ ρ satisfying E ( ρ ) = inf ρ ∈A +¯ ρ E ( ρ ) .Furthermore, ρ (cid:54)≡ ¯ ρ , ρ > , ρ ∈ C / ( R ) ∩ L ∞ ( R ) and ρ ( x ) → ¯ ρ as | x | → ∞ . One would naturally expect minimizers in Theorem 3.2 to coincide with the onein Theorem 3.1 in many situations, yet it seems difficult to prove this at themoment. It is clear, however, that ρ from Theorem 3.2 is a local minimizer of E with respect to smooth perturbations with compact support. As a consequence,these minimizers solve pointwise the Euler-Lagrange equation associated withthe energy, which for ρ > a ( − ∆ ) / ( √ ρ ) + √ ρ (cid:0) √ ρ − √ ¯ ρ + V + bU (cid:1) . (3.10)We also note that the assumption of boundedness of V in Theorem 3.2 is neededto ensure strict positivity of the minimizer, which is required to obtain (3.10). Inaddition, positivity of ρ implies further regularity under additional smoothnessassumptions on V . In particular, ρ ∈ C ∞ ( R ) if V ∈ C ∞ ( R ), see Remark 7.1.We note that one of the main differences with the result of Theorem 3.1 inthe case of Theorem 3.2 is that there is uniqueness of minimizers, which is dueto a kind of strict convexity of the functional E over A +¯ ρ . In fact, due to thisstrict convexity one should further expect uniqueness of solutions of (3.10) and,in particular, that the minimizer ρ in Theorem 3.2 is radially-symmetric, if sois the potential V [40]. rbital-free density functional theory of out-of-plane charge screening in graphene 13 Remark 3.1.
It is easy to see from (3.5) that if V ≡
0, the unique minimizer of E over A ¯ ρ is ρ = ¯ ρ . At the same time, if ¯ ρ > ρ = ¯ ρ is a minimizer of E over A ¯ ρ , by (3.10) we have V ≡ A +¯ ρ ⊂ A ¯ ρ also implies that if ¯ ρ > V ∈ ˚ H / ( R ) and V (cid:54)≡ ρ ∈A ¯ ρ E ( ρ ) < E over A ¯ ρ are positive, in the case of ¯ ρ > V which are, in some sense, “small”. The smallness ofthe potential is expressed in terms of the magnitude of its ˚ H / ( R ) norm. Ourresult is given by the following theorem. Theorem 3.3.
Let ¯ ρ > , let E be defined by (3.5) with V ∈ ˚ H / ( R ) and let V (cid:54)≡ . Then there exists a constant C > depending only on a , b and ¯ ρ suchthat if (cid:107) V (cid:107) ˚ H / ( R ) ≤ C , then the unique minimizer ρ > of E over A +¯ ρ inTheorem 3.2 coincides with the minimizer of E over A ¯ ρ in Theorem 3.1. We note that in the parameter regime of Theorem 3.3 the minimizer ρ > ρ >
0. In particular, if (cid:107) V (cid:107) ˚ H / ( R ) →
0, one expects torecover, to the leading order, the solution of (3.10) linearized around ρ = ¯ ρ ,which expresses the linear response of the system to the perturbation by thepotential V and describes screening of the external charge by free electrons in thegraphene layer. A more detailed analysis of this phenomenon will be carried outin the forthcoming paper [40]. Note that within the Thomas-Fermi type modelsof the usual electron systems screening was studied mathematically in [38] for theThomas-Fermi model and in [10] for the Thomas-Fermi-von Weizs¨acker model.We now focus on the main situation of physical interest, in which the layer isat the neutrality point. In particular, we wish to investigate how a graphene layerreacts to external charges in the presence of a supply of electrons from a lead atinfinity. Fixing ¯ ρ = 0, we know that under the assumptions of Theorem 3.1 thereis a non-trivial minimizer in the class A . As we already mentioned, we do notknow whether this minimizer also belongs to A +0 , even for a potential defined in(1.3) with a positive measure µ . Nevertheless, if we restrict the admissible classto A +0 , we have the following analog of Theorem 3.2. Theorem 3.4.
Let ¯ ρ = 0 , let E be defined by (3.5) with V ∈ ˚ H / ( R ) , andlet inf ρ ∈A +¯ ρ E ( ρ ) < . Then there exists a unique ρ ∈ A +¯ ρ satisfying E ( ρ ) =inf ρ ∈A +¯ ρ E ( ρ ) . Furthermore, ρ > , ρ ∈ C / ( R ) ∩ L ∞ ( R ) and ρ ( x ) → as | x | → ∞ . Let us point out that, in contrast to Theorem 3.2, the condition that V (cid:54)≡ (cid:107) V (cid:107) ˚ H / ( R ) the energy E in (1.4) cannot have non-trivial minimizers. We illustrate this point by considering the case of the energy E in (1.2), which is also of particular interest because of its physical significance.Defining a c := Γ ( )2 Γ ( ) , (3.11)where Γ ( x ) is the Gamma function and a c ≈ . E in (1.2). Theorem 3.5.
Let ¯ ρ = 0 and let E be defined by (3.5) with V ( x ) = − | x | ) / . (3.12) Then:(i) If a ≥ a c , then ρ = 0 is the unique minimizer of E over A .(ii) If a < a c , then there exists a minimizer ρ (cid:54)≡ of E over A +0 . Thus, for a sufficiently large (or, equivalently, for the impurity valence Z suffi-ciently small or the effective dielectric constant (cid:15) d sufficiently large, see (2.10))there can be no bound states between the charge carriers in graphene and a singlecharged impurity. In other words, this implies a surprising result that for a ≥ a c the charged impurity elicits no response from the electrons in the graphene layer(within the considered density functional theory). The bifurcation at a = a c is determined by a fine balance between the first term in the energy and thepotential term, which has the same asymptotics when | x | → ∞ as the Hardypotential for ( − ∆ ) / .Note that the statement of Theorem 3.5 obviously remains true if A +0 isreplaced with A . Also note that the magnitude of b does not play any role forexistence vs. non-existence of non-trivial minimizers in this case. At the sametime, as we will show in the forthcoming paper [40], both the values of a and b , together with the (finite) L norm of the minimizer ρ ∈ A +0 determine thealgebraic rate of decay of ρ ( x ) as | x | → ∞ . Specifically, we expect ρ ( x ) ∼ | x | s , | x | → ∞ , (3.13)where s ∈ (1 ,
2) is the unique solution of the algebraic equation2 aΓ ( s +12 ) Γ ( − s ) Γ ( − s ) Γ ( s ) = 1 − b π (cid:107) ρ ( x ) (cid:107) L ( R ) , (3.14)which is formally obtained by linearizing (3.10) with respect to √ ρ , using theleading order asymptotics of V and U in the far field and looking for distribu-tional solutions in the form appearing in (3.13). This prediction is confirmed bythe results of the numerical solution of (3.10). Figure 1 shows the solution of(3.10) for a = 1 and b = 1 (we refer to [40] for further details), for which we rbital-free density functional theory of out-of-plane charge screening in graphene 15 (cid:200) x (cid:200) Ρ a (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:200) x (cid:200) Ρ (cid:126) (cid:200) x (cid:45) b (cid:76) Fig. 1.
The minimizer ρ in Theorem 3.5 for a = 1 and b = 1, plotted on a linear (a) andlogarithmic (b) scale. found (cid:107) ρ (cid:107) L ( R ) (cid:39) .
95 and ρ ( x ) (cid:39) . | x | − . for | x | (cid:29)
1. This agrees wellwith (3.14). Thus, in contrast to previous studies, our model predicts a non-trivial dependence of the algebraic decay rate of the positive mininimizers onthe parameters. Note that since for s ∈ (1 ,
2) the term multiplying a in (3.14)is negative, we have (cid:107) ρ ( x ) (cid:107) L ( R ) > πb − . In the original physical variablesit means that the total charge induced in the graphene layer exceeds in abso-lute value the external out-of-plane charge. Note that this is similar to what isobserved in the Thomas-Fermi-von Weizs¨acker model of a single atom [6].
4. Preliminaries
Recall that the homogeneous Sobolev space ˚ H / ( R )can be defined as the completion of C ∞ c ( R ) with respect to the Gagliardo’snorm (cid:107) u (cid:107) H / ( R ) := 14 π (cid:90) (cid:90) R × R | u ( x ) − u ( y ) | | x − y | d x d y. (4.1)By Plancherel’s identity (cf. [24, Lemma 3.1]), on C ∞ c ( R ) the (cid:107) · (cid:107) ˚ H / ( R ) –normadmits an equivalent Fourier representation (cid:107) u (cid:107) H / ( R ) = (cid:90) R | | k | / ˆ u k | d k (2 π ) , ˆ u k = (cid:90) R e i k · x u ( x ) d x, (4.2)which suggests the notation (cid:107) u (cid:107) H / ( R ) =: (cid:90) R |∇ u ( x ) | d x, (4.3)which we often use in this paper. By the fractional Sobolev inequality [36, The-orem 8.4], [15, Theorem 6.5], (cid:107) u (cid:107) H / ( R ) ≥ √ π (cid:107) u (cid:107) L ( R ) , ∀ u ∈ C ∞ c ( R ) . (4.4) In particular, the space ˚ H / ( R ) is a well-defined space of functions and˚ H / ( R ) ⊂ L ( R ) . (4.5)The space ˚ H / ( R ) is also a Hilbert space, with the scalar product associatedto (4.1) given by (cid:104) u, v (cid:105) ˚ H / ( R ) := 14 π (cid:90) (cid:90) R × R ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | d x d y. (4.6)The dual space to ˚ H / ( R ) is denoted ˚ H − / ( R ). According to the Riesz rep-resentation theorem, for every F ∈ ˚ H − / ( R ) there exists a uniquely defined potential v ∈ ˚ H / ( R ) such that (cid:104) v, ϕ (cid:105) ˚ H / ( R ) = (cid:104) F, ϕ (cid:105) ∀ ϕ ∈ ˚ H / ( R ) , (4.7)where (cid:104) F, ·(cid:105) : ˚ H / ( R ) → R denotes the bounded linear functional generatedby F . Moreover, (cid:107) v (cid:107) ˚ H / ( R ) = (cid:107) F (cid:107) ˚ H − / ( R ) , (4.8)so the duality (4.7) is an isometry. The potential v ∈ ˚ H / ( R ) satisfying (4.7)is interpreted as the weak solution of the linear equation( − ∆ ) / v = F in R . (4.9)Recall that for functions u ∈ C ∞ ( R ) ∩ L ( R , (1 + | x | ) − d x ), the fractionalLaplacian ( − ∆ ) / can be defined as( − ∆ ) / u ( x ) = 14 π (cid:90) R u ( x ) − u ( x + y ) − u ( x − y ) | y | d y ( x ∈ R ) . (4.10)Note that the second order Taylor expansion of function u yields that the strongsingularity of the integrand at the origin is removed, and (4.10) can be under-stood as a converging Lebesgue integral, see [15, Lemma 3.2]. Of course, theweighted second order differential quotient in (4.10) coincides with a more stan-dard definition of ( − ∆ ) / as a pseudodifferential operator, in the sense that forall u ∈ C ∞ c ( R ), (cid:92) (cid:0) ( − ∆ ) / u (cid:1) k = | k | ˆ u k , (4.11)cf. [15, Proposition 3.3]. In particular, this makes the definition of ( − ∆ ) / in(4.10) consistent with the notation used in (4.9).Note that if u ∈ C ∞ c ( R ) then ( − ∆ ) / u ∈ C ∞ ( R ), but is not compactlysupported and in fact,( − ∆ ) / u = O ( | x | − ) as | x | → ∞ , (4.12) rbital-free density functional theory of out-of-plane charge screening in graphene 17 see [42, Lemma 1.2]. In particular, this shows that the operator ( − ∆ ) / couldbe extended by duality to the weighted space L ( R , (1 + | x | ) − d x ), that is for u ∈ L ( R , (1 + | x | ) − d x ), (cid:104) ( − ∆ ) / u, ϕ (cid:105) = (cid:90) R u ( x )( − ∆ ) / ϕ ( x ) d x ∀ ϕ ∈ C ∞ c ( R ) (4.13)and this definition agrees with (4.10) in the case u ∈ C ∞ c ( R ), see [51, p. 73].Clearly, ˚ H / ( R ) ⊂ L ( R , (1 + | x | ) − d x ). In particular, this implies that for v ∈ ˚ H / ( R ), (cid:104) v, ϕ (cid:105) ˚ H / ( R ) = (cid:90) R v ( x )( − ∆ ) / ϕ ( x ) d x ∀ ϕ ∈ C ∞ c ( R ) . (4.14)When f ∈ C ∞ c ( R ), the left inverse to ( − ∆ ) / is represented by the Rieszpotential , i.e., if u is the weak solution of ( − ∆ ) / u = f then u admits theintegral representation u ( x ) = ( − ∆ ) − / f ( x ) = 12 π (cid:90) R f ( y ) | x − y | d y, (4.15)see [42, Lemma 1.3]. Such integral representation could be extended to a widerclass of functions and (signed) measures, cf. [42, Lemma 1.8, 1.11]. In particular,taking f = δ ( x ), we obtain that 1 / (2 π | x | ) is the fundamental solution of ( − ∆ ) / .We emphasize, however, that not every potential of a linear functional f ∈ ˚ H − / ( R ) admits an integral representation (4.15). Similarly, not every linearfunctional f ∈ ˚ H − / ( R ) admits an integral representation of the norm in termsof the Coulomb energy. If f ∈ L loc ( R ) satisfies (cid:90) (cid:90) R × R | f ( x ) || f ( y ) || x − y | d x d y < + ∞ . (4.16)then f ∈ ˚ H − / ( R ) in the sense that (cid:104) f, ϕ (cid:105) := (cid:90) R f ( x ) ϕ ( x ) d x (4.17)is a bounded linear functional on ˚ H / ( R ) and the norm of (cid:104) f, ·(cid:105) is expressedin terms of the Coulomb energy (cid:107) f (cid:107) H − / ( R ) = 12 π (cid:90) (cid:90) R × R f ( x ) f ( y ) | x − y | d x d y, (4.18)see e.g. [42, pp. 96-97]. In particular, from Sobolev inequality (4.4) we concludeby duality that L / ( R ) ⊂ ˚ H − / ( R ) (4.19)and (4.18) is valid for every f ∈ L / ( R ). But at the same time, one couldconstruct a sequence of sign–changing functions { f n } ⊂ C ∞ c ( R ) such that { f n } is a Cauchy sequence in ˚ H − / ( R ), but { f n } does not converge a.e. to a mea-surable function or more generally, to a (signed) measure on R . See [4, 47] or[33, Theorem 1.19], [17, p. 97] for other relevant examples which go back to H.Cartan [12, Remark 13 on p. 87]. Below we present a different example whichinvolves smooth functions, rather than measures like in Cartan’s type examples. Example 4.1.
Define u a ( x , x ) = a / exp( −| x | ) cos( ax ) . (4.20)Then, using Fourier transform, we can calculate that (cid:107) u a (cid:107) H − / ( R ) = √ π / ae − a (cid:16) e a I (cid:0) a (cid:1) + 1 (cid:17) , (4.21)where I ( z ) is the modified Bessel function of the first kind. Taking the limit a → ∞ , one gets lim a →∞ || u a || H − / ( R ) = π . (4.22)A Cauchy sequence in ˚ H − / ( R ) that fails to converge to a signed measure canthen be constructed as u n ( x , x ) = n (cid:88) k =1 e k/ exp( −| x | ) cos( e k x ) . (4.23)Since this series is dominated in ˚ H − / ( R ) by a geometric series, it convergesin ˚ H − / ( R ). But clearly it does not converge to a signed measure. We recall the well-knownHardy–Littlewood–Sobolev [53, Theorem 1 in Section V.1.2] and H¨older esti-mates on the Riesz potentials of functions in L p ( R ). Surprisingly, we were notable to find a concise reference to H¨older estimate, although the result is stan-dard. Instead, we refer to [25, Theorem 5.2], where the estimate is obtained inan abstract framework of fractional integral operators. Lemma 4.1.
Let f ∈ L s ( R ) for some s ∈ (1 , and v ( x ) = 12 π (cid:90) R f ( y ) | x − y | d y ( x ∈ R ) . (4.24) Then v ∈ L t ( R ) with t = s − and (cid:107) v (cid:107) L t ( R ) ≤ C (cid:107) f (cid:107) L s ( R ) , (4.25) for some C > depending only on s . Furthermore, if f ∈ L s ( R ) ∩ L ( R , (1 + | x | ) − d x ) for some s > , then v ∈ L ∞ ( R ) ∩ C − s ( R ) and | v ( x ) − v ( y ) | ≤ C (cid:107) f (cid:107) L s ( R ) | x − y | − s ∀ x, y ∈ R , (4.26) for some C > depending only on s . rbital-free density functional theory of out-of-plane charge screening in graphene 19 Remark 4.1.
The assumption f ∈ L ( R , (1+ | x | ) − d x ) in the second part of thelemma is a necessary and sufficient condition which ensures that | v ( x ) | < + ∞ a.e. in R , assuming that the operator in (4.24) is understood in the (Lebesgue)integral sense, c.f. [33, (1.3.10) on p. 61]. Observe that by H¨older inequality allthe assumptions of the second part of Lemma 4.1 are satisfied, if f ∈ L s ( R ) forall s ∈ [ s , s ] for some 1 < s < < s < ∞ . We are going to show that although ( − ∆ ) / is a non-local operator, the interior regularity of solutions of (4.9) does not depend onthe behavior of the right-hand side at infinity. The proof of this basic fact canbe found in [51, Proposition 2.22]. Here, however, we give a quantitative versionof the above statement. Lemma 4.2.
Let f ∈ L loc ( R ) , let p ≥ and let u ∈ L p ( R ) be such that (cid:104) u, ϕ (cid:105) ˚ H / ( R ) = (cid:90) R f ( x ) ϕ ( x ) d x ∀ ϕ ∈ C ∞ c ( R ) . (4.27) Assume that f = 0 on B R (0) for some R > . Then u ∈ C ∞ ( ¯ B R (0)) and forevery n ≥ (cid:107)∇ n u (cid:107) L ∞ ( B R (0)) ≤ CR − n − p (cid:107) u (cid:107) L p ( R ) (4.28) for some C > depending only on n and p .Proof. Let η R ( x ) = η ( | x | /R ), where η ∈ C ∞ ( R ) is a smooth cut-off functionsuch that η ( x ) = 1 for all | x | > η ( x ) = 0 for all | x | < , and 0 ≤ η ≤ ϕ ∈ C ∞ c ( R ) supported on B R (0), let ψ ∈ ˚ H / ( R ) be a weak solutionof ( − ∆ ) / ψ = ϕ . By (4.15) we have ψ ( x ) = 12 π (cid:90) R ϕ ( y ) | x − y | d y, (4.29)and, in particular, ψ ∈ C ∞ ( R ). Then (1 − η R ) ψ ∈ C ∞ c ( R ) and is supported on B R (0). Testing (4.27) with (1 − η R ) ψ and taking into account (4.14), we obtain0 = (cid:104) u, (1 − η R ) ψ (cid:105) ˚ H / ( R ) = (cid:90) R u ( x )( − ∆ ) / (cid:16) (1 − η R ) ψ (cid:17) ( x ) d x = (cid:90) B R (0) u ( x ) ϕ ( x ) d x − (cid:90) R u ( x )( − ∆ ) / ( η R ψ )( x ) d x. (4.30) Inserting the definition of ( − ∆ ) / from (4.10) and changing the order ofintegration in the last integral in (4.30) yields (cid:90) R u ( x )( − ∆ ) / ( η R ψ )( x ) d x = 18 π (cid:90) R u ( x ) (cid:90) R | y | − (cid:90) B R (0) (cid:18) η R ( x ) | x − z | − η R ( x + y ) | x + y − z |− η R ( x − y ) | x − y − z | (cid:19) ϕ ( z ) d z d y d x = 18 π (cid:90) B R (0) (cid:90) R (cid:18)(cid:90) R | y | − (cid:18) η R ( x ) | x − z | − η R ( x + y ) | x + y − z |− η R ( x − y ) | x − y − z | (cid:19) d y (cid:19) u ( x ) ϕ ( z ) d x d z = (cid:90) B R (0) (cid:90) R J R ( x, z ) u ( x ) ϕ ( z ) d x d z, (4.31)where for x ∈ R and z ∈ B R (0) we introduced J R ( x, z ) := 18 π (cid:90) R | y | − (cid:18) η R ( x ) | x − z | − η R ( x + y ) | x + y − z | − η R ( x − y ) | x − y − z | (cid:19) d y. (4.32)Observe that J R ( x, z ) = ( − ∆ ) / x j R ( x, z ) , j R ( x, z ) := 12 π η R ( x ) | x − z | . (4.33)Clearly, j R ( x, z ) = 0 for x ∈ B R/ (0) and j R ∈ C ∞ ( R × ¯ B R (0)), with |∇ nz j R ( x, z ) | ≤ c n ( R + | x − z | ) − ( n +1) , (4.34) |∇ x ∇ nz j R ( x, z ) | ≤ c n R − ( R + | x − z | ) − ( n +1) (4.35)for all n ≥ c n > n and the choice of η ). Then |∇ nz J R ( x, z ) | ≤ π (cid:90) R | y | − (cid:12)(cid:12)(cid:12) ∇ nz j R ( x, z ) − ∇ nz j R ( x + y, z ) − ∇ nz j R ( x − y, z ) (cid:12)(cid:12)(cid:12) d y = 18 π (cid:90) B R (0) | y | − (cid:12)(cid:12)(cid:12) ∇ nz j R ( x, z ) − ∇ nz j R ( x + y, z ) − ∇ nz j R ( x − y, z ) (cid:12)(cid:12)(cid:12) d y + 18 π (cid:90) R \ B R (0) | y | − (cid:12)(cid:12)(cid:12) ∇ nz j R ( x, z ) − ∇ nz j R ( x + y, z ) − ∇ nz j R ( x − y, z ) (cid:12)(cid:12)(cid:12) d y ≤ π (cid:107)∇ x ∇ nz j R ( · , z ) (cid:107) L ∞ ( R ) (cid:90) B R (0) | y | − d y + 12 π (cid:107)∇ nz j R ( · , z ) (cid:107) L ∞ ( R ) (cid:90) R \ B R (0) | y | − d y ≤ C n R − n − , (4.36) rbital-free density functional theory of out-of-plane charge screening in graphene 21 for some C n >
0. In particular, for any x ∈ R , J R ( x, · ) ∈ C ∞ ( ¯ B R (0)).We next prove that for some c n > |∇ nz J R ( x, z ) | ≤ c n R n − ( R + | x | ) ∀ x ∈ R , ∀ z ∈ ¯ B R (0) . (4.37)For | x | ≤ R , the estimate follows from (4.36). Now assume | x | ≥ R . Then η R ( x ) = 1, and since 1 / (2 π | x | ) is the fundamental solution for ( − ∆ ) / , wehave for | z | ≤ R π (cid:90) R | y | − (cid:18) | x − z | − | x + y − z | − | x − y − z | (cid:19) d y = 0 . (4.38)Using this fact we can rewrite J R ( x, z ) = 18 π (cid:90) R − η R ( x − y ) | x − y − z | | y | − d y + 18 π (cid:90) R − η R ( x + y ) | x + y − z | | y | − d y = 18 π (cid:90) R | y − z | (cid:32) − η R ( y ) | x − y | + 1 − η R ( y ) | x + y | (cid:33) d y =: 18 π (cid:90) R | y − z | h R ( x, y ) d y. (4.39)Notice that for fixed x with | x | ≥ R , h R ( x, · ) ∈ C ∞ c ( R ) and its support iscontained in B R (0). Therefore, |∇ nz J R ( x, z ) | ≤ π (cid:90) B R (0) | y − z | (cid:12)(cid:12) ∇ ny h R ( x, y ) (cid:12)(cid:12) d y. (4.40)For y ∈ B R (0) and | x | ≥ R , we have the estimate (cid:12)(cid:12) ∇ ny h R ( x, y ) (cid:12)(cid:12) ≤ C n R − n | x | − for some C n >
0. Therefore, |∇ nz J R ( x, z ) | ≤ C n R n | x | (cid:90) B R (0) | y − z | d y ≤ C (cid:48) n R n − | x | , (4.41)for some C (cid:48) n > z ∈ ¯ B R (0) we have u ( z ) = (cid:90) R J R ( x, z ) u ( x ) d x, (4.42)and, since (4.37) leads to (cid:107)∇ nz J R ( · , z ) (cid:107) L pp − ( R ) ≤ CR − n − /p for some C > n , p and the choice of η , the statement of the lemma followsby H¨older inequality. (cid:117)(cid:116) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) u (cid:70) a (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) u S b (cid:76) Fig. 2. (a) Plot of Φ ( u ) and (b) plot of S ( u ) for ¯ u = 1.
5. Variational setting
Recall that for a given ρ ∈ A ¯ ρ we define u by u := sgn( ρ ) (cid:112) | ρ | − sgn(¯ ρ ) (cid:112) | ¯ ρ | (5.1)and set ¯ u := (cid:112) | ¯ ρ | sgn ¯ ρ . Then u ∈ ˚ H / ( R ) in view of the definition of A ¯ ρ .Since sgn( ρ ) (cid:112) | ρ | = u + ¯ u , we can define (cid:90) R (cid:12)(cid:12) ∇ (cid:0)(cid:112) | ρ ( x ) | sgn( ρ ( x )) (cid:1)(cid:12)(cid:12) d x :=14 π (cid:90) (cid:90) R × R | ( u ( x ) + ¯ u ) − ( u ( y ) + ¯ u ) | | x − y | d x d y = (cid:107) u (cid:107) H / ( R ) , (5.2)which justifies and clarifies the notation used in Sections 1–3 of the paper.Throughout the rest of the paper we assume, without loss of generality, that¯ ρ ≥
0, and, hence, ¯ u ≥ S ( u ) := | u + ¯ u | ( u + ¯ u ) − | ¯ u | ¯ u = (cid:40) uu + u , u ≥ − ¯ u, − u − uu − u , u < − ¯ u, (5.3)and Φ ( u ) := 23 ( | u + ¯ u | − | ¯ u | ) − ¯ uS ( u ) = (cid:40) u + ¯ uu , u ≥ − ¯ u, − u − ¯ uu + ¯ u , u < − ¯ u. (5.4)The graphs of Φ ( u ) and S ( u ) for ¯ u = 1 are presented in Fig. 2. Clearly S, Φ ∈ C ( R ) and both functions are smooth functions of u ∈ R except at u = − ¯ u .Moreover c (¯ u | u | + | u | ) ≤ Φ ( u ) ≤ C (¯ u | u | + | u | ) ( u ∈ R ) , (5.5) c (¯ u | u | + | u | ) ≤ S ( u ) sgn( u ) ≤ C (¯ u | u | + | u | ) ( u ∈ R ) , (5.6) rbital-free density functional theory of out-of-plane charge screening in graphene 23 for some universal C > c >
0. Therefore, for u ∈ C ∞ c ( R ), the energy E ( u ) canbe written as (with a slight abuse of notation, we use the same letter to denoteboth the energy as a function of ρ and that as a function of u in the rest of thepaper) E ( u ) = a (cid:107) u (cid:107) H / ( R ) + (cid:90) R Φ ( u ( x )) d x + (cid:90) R V ( x ) S ( u ( x )) d x + b (cid:90) (cid:90) R × R S ( u ( x )) S ( u ( y )) | x − y | d x d y. (5.7)Given u ∈ ˚ H / ( R ), (4.5) and (5.6) imply that S ( u ) ∈ L loc ( R ). Then for all ϕ ∈ C ∞ c ( R ) we can define (cid:104) S ( u ) , ϕ (cid:105) := (cid:90) R S ( u ( x )) ϕ ( x ) d x. (5.8)We say S ( u ) ∈ ˚ H − / ( R ), if the linear functional (cid:104) S ( u ) , ·(cid:105) defined in (5.8) isbounded by a multiple of (cid:107) ϕ (cid:107) ˚ H / ( R ) . In that case (cid:104) S ( u ) , ·(cid:105) is understood as theunique continuous extension of (5.8) to ˚ H / ( R ). Note that S ( u ) ∈ ˚ H − / ( R )does not necessarily imply that S ( u ) w ∈ L ( R ) for every w ∈ ˚ H / ( R ). In otherwords, (cid:104) S ( u ) , ·(cid:105) does not always admit an integral representation on ˚ H / ( R ),as observed by Brezis and Browder in [9] in the context of H ( R N ). H . Introduce the class H := (cid:110) u ∈ ˚ H / ( R ) : S ( u ) ∈ ˚ H − / ( R ) (cid:111) . (5.9)As discussed in Section 5.1, this is an equivalent way of writing the class A ¯ ρ .Given u ∈ H , Riesz’s representation theorem uniquely defines a potential U S ( u ) ∈ ˚ H / ( R ) such that (cid:104) U S ( u ) , ϕ (cid:105) ˚ H / ( R ) = (cid:104) S ( u ) , ϕ (cid:105) ∀ ϕ ∈ ˚ H / ( R ) . (5.10)In particular, from the Sobolev embedding (4.5) combined with (5.5) we obtainthe following inclusions: { u ∈ H : E ( u ) < + ∞} ⊂ L ( R ) ∩ L ( R ) if ¯ u (cid:54) = 0, (5.11) { u ∈ H : E ( u ) < + ∞} ⊂ L ( R ) ∩ L ( R ) if ¯ u = 0 . (5.12) Remark 5.1.
In fact, using a fractional extension of the Brezis-Browder argumentin [9], one can establish stronger inclusions:
H ⊂ L ( R ) ∩ L ( R ) if ¯ u (cid:54) = 0, (5.13) H ⊂ L ( R ) ∩ L ( R ) if ¯ u = 0 . (5.14) We refer to the forthcoming work [40] for the details. Moreover, these inclu-sions are, in some sense optimal. To see the optimality of (5.13), choose u ∈ C ∞ c ( B (0)), a vector e ∈ R with | e | = 1 and for N ∈ N let u N ( x ) := 1 √ N N (cid:88) k =1 u (cid:0) x + k exp( N ) e (cid:1) . (5.15)It is standard to check (cf. (4.12) for the ˚ H / –term and [48, p. 363] for theCoulomb term) that (cid:107) u N (cid:107) ˚ H / ( R ) (cid:39) (cid:107) S ( u N ) (cid:107) ˚ H − / ( R ) (cid:39) C, (5.16)while (cid:107) u N (cid:107) L p ( R ) = O ( N p − ) . (5.17)We conclude that the sequence { u N } is not bounded in L p ( R ) for any p < u N , similar to those in [48, Proof of Theorem 1.5].
6. Proof of Theorems 3.1 and 3.3 If V ∈ ˚ H / ( R ) then we can rewrite E in termsof u and the associated potential U S ( u ) as E ( u ) = a (cid:107) u (cid:107) H / ( R ) + (cid:90) R Φ ( u ( x )) d x + (cid:104) V, U S ( u ) (cid:105) ˚ H / ( R ) + b (cid:107) U S ( u ) (cid:107) H / ( R ) . (6.1)In particular, it is easy to see that − b (cid:107) V (cid:107) H / ( R ) ≤ inf u ∈H E ( u ) ≤ . (6.2)We are going to prove that E attains a minimizer on H . Proposition 6.1. If V ∈ ˚ H / ( R ) then there exists u ∈ H such that E ( u ) =inf u ∈H E ( u ) .Proof. Consider a minimizing sequence { u n } ⊂ H and the corresponding se-quence of potentials { U S ( u n ) } ⊂ ˚ H / ( R ) from (5.10). Clearly,sup n (cid:107) u n (cid:107) H / ( R ) ≤ C, (6.3)sup n (cid:107) U S ( u n ) (cid:107) H / ( R ) ≤ C, (6.4)Hence, we may extract subsequences, still denoted by { u n } and { U S ( u n ) } suchthat u n (cid:42) u in ˚ H / ( R ) , (6.5) U S ( u n ) (cid:42) v in ˚ H / ( R ) , (6.6) rbital-free density functional theory of out-of-plane charge screening in graphene 25 for some u , v ∈ ˚ H / ( R ). Using a fractional version of Rellich-Kondrachovtheorem [15, Corollary 7.2], we conclude that u n → u in L ploc ( R ) for all 1 ≤ p < , (6.7)and, upon extraction of another subsequence, that u n ( x ) → u ( x ) for a.e. x ∈ R . Using (6.7), (5.6) and strong continuity of S as a Nemytskii operator from L ploc ( R ) into L qloc ( R ) with q ≤ p/ S ( u n ) → S ( u ) in L qloc ( R ) for all 1 ≤ q < . (6.8)Using (5.10), (5.8) and (6.8), similarly to an argument in the proof of [48, Propo-sition 2.4], for every fixed ϕ ∈ C ∞ c ( R ) we obtain (cid:104) v , ϕ (cid:105) ˚ H / ( R ) ← (cid:104) U S ( u n ) , ϕ (cid:105) ˚ H / ( R ) = (cid:104) S ( u n ) , ϕ (cid:105) = (cid:90) R S ( u n ( x )) ϕ ( x ) d x → (cid:90) R S ( u ( x )) ϕ ( x ) d x. (6.9)Therefore, (cid:104) v , ϕ (cid:105) ˚ H / ( R ) = (cid:90) R S ( u ( x )) ϕ ( x ) d x, ∀ ϕ ∈ C ∞ c ( R ) . (6.10)Note that (cid:104) v , ·(cid:105) ˚ H / ( R ) is a bounded linear functional on ˚ H / ( R ), since v ∈ ˚ H / ( R ). Therefore S ( u ) ∈ ˚ H − / ( R ). In particular, this means that u ∈ H and v = U S ( u ) . (6.11)We conclude that E ( u ) = a (cid:107) u (cid:107) H / ( R ) + (cid:90) R Φ ( u ( x )) d x + (cid:104) V, U S ( u ) (cid:105) ˚ H / ( R ) + b (cid:107) U S ( u ) (cid:107) H / ( R ) ≤ lim inf n →∞ E ( u n ) . (6.12)This follows from the weak lower semicontinuity of the norm (cid:107) · (cid:107) ˚ H / ( R ) , con-tinuity of the linear functional (cid:104) V, ·(cid:105) ˚ H / ( R ) on ˚ H / ( R ), and from the non-negativity of the function Φ which allows to apply Fatou lemma in the integralterm which contains Φ . (cid:117)(cid:116) In order to derive the Euler–Lagrange equationfor E , we first establish three auxiliary lemmas. Lemma 6.1.
Let u ∈ H and h ∈ C ∞ c ( R ) . Then u + th ∈ H for every t ∈ R . Proof.
Since obviously u + th ∈ ˚ H / ( R ), it remains to prove that S ( u + th ) ∈ ˚ H − / ( R ). Consider F ( x ) := S ( u ( x ) + th ( x )) − S ( u ( x )). Clearly, F has com-pact support, and by (5.6) we have F ∈ L ( R ). Therefore, we also have F ∈ L / ( R ), and, hence, by (4.19) the functional (cid:104) F, ϕ (cid:105) := (cid:90) R ( S ( u ( x ) + th ( x )) − S ( u ( x ))) ϕ ( x ) d x ( ϕ ∈ C ∞ c ( R )) (6.13)can be continuously extended to the whole of ˚ H / ( R ). Thus S ( u + th ) − S ( u ) ∈ ˚ H − / ( R ), and since S ( u ) ∈ ˚ H − / ( R ) by assumption, this completes theproof. (cid:117)(cid:116) Lemma 6.2.
Let u ∈ H and h ∈ C ∞ c ( R ) . Then S (cid:48) ( u ) h ∈ ˚ H − / ( R ) ∩ L ( R ) ,and for every ϕ ∈ ˚ H / ( R ) , lim t → t (cid:104) S ( u + th ) − S ( u ) , ϕ (cid:105) = (cid:90) R S (cid:48) ( u ( x )) h ( x ) ϕ ( x ) d x. (6.14) Proof.
Note that S (cid:48) ( u ) = 2 | u + ¯ u | , (6.15)and, hence, S (cid:48) ( u ) ∈ L loc ( R ) by (4.5). Therefore, in view of the fact that h ∈ C ∞ c ( R ), we have S (cid:48) ( u ) h ∈ L ( R ) ∩ L / ( R ) and, again, by (4.19), this impliesthat S (cid:48) ( u ) h ∈ ˚ H − / ( R ).At the same time, by the argument in the proof of Lemma 6.1 we have anintegral representation (cid:104) S ( u + th ) − S ( u ) , ϕ (cid:105) = (cid:90) R (cid:0) S ( u ( x ) + th ( x )) − S ( u ( x )) (cid:1) ϕ ( x ) d x (6.16)for every ϕ ∈ ˚ H / ( R ). Using (6.16) and the mean value theorem, for some θ ( t, · ) ∈ L ∞ ( R ) with (cid:107) θ ( t, · ) (cid:107) L ∞ ≤
1, we obtain1 t (cid:104) S ( u + th ) − S ( u ) , ϕ (cid:105) = 1 t (cid:90) R (cid:0) S ( u ( x ) + th ( x )) − S ( u ( x )) (cid:1) ϕ ( x ) d x = (cid:90) R S (cid:48) (cid:0) u ( x ) + tθ ( t, x ) h ( x ) (cid:1) h ( x ) ϕ ( x ) d x, (6.17)where the latter integral converges, since S (cid:48) ( u + tθ ( t, · ) h ) ∈ L loc ( R ) in view of(6.15) and (4.5). Then (6.14) follows by the Lebesgue dominated convergence. (cid:117)(cid:116) Lemma 6.3.
Let u ∈ H and h ∈ C ∞ c ( R ) . Then lim t → t (cid:16) (cid:107) U S ( u + th ) (cid:107) H / ( R ) − (cid:107) U S ( u ) (cid:107) H / ( R ) (cid:17) = 2 (cid:90) R U S ( u ) ( x ) S (cid:48) ( u ( x )) h ( x ) d x. (6.18) rbital-free density functional theory of out-of-plane charge screening in graphene 27 Proof.
Since S ( u + th ) ∈ ˚ H − / ( R ) by Lemma 6.1, the potential U S ( u + ht ) ∈ ˚ H / ( R ) is well-defined. Then using (5.10) we obtain (cid:107) U S ( u + th ) (cid:107) H / ( R ) − (cid:107) U S ( u ) (cid:107) H / ( R ) = 2 (cid:104) S ( u + th ) − S ( u ) , U S ( u ) (cid:105) + (cid:104) S ( u + th ) − S ( u ) , U S ( u + th ) − S ( u ) (cid:105) . (6.19)Similarly to (6.17), for some θ ( t, · ) ∈ L ∞ ( R ) with (cid:107) θ ( t, · ) (cid:107) L ∞ ≤ t (cid:12)(cid:12) (cid:104) S ( u + th ) − S ( u ) , U S ( u + th ) − S ( u ) (cid:105) (cid:12)(cid:12) = 1 t (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R (cid:0) S ( u ( x ) + th ( x )) − S ( u ( x )) (cid:1) U S ( u + th ) − S ( u ) ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107) S (cid:48) ( u + tθ ( t, · ) h ) h (cid:107) L / ( R ) (cid:107) U S ( u + th ) − S ( u ) (cid:107) ˚ H / ( R ) , (6.20)for some C > t . Since h is compactly supported, by Lebesguedominated convergence we conclude that (cid:107) S (cid:48) ( u + tθ ( t, · ) h ) h (cid:107) L / ( R ) → (cid:107) S (cid:48) ( u ) h (cid:107) L / ( R ) as t → , (6.21) (cid:107) S ( u + th ) − S ( u ) (cid:107) L / ( R ) → t → . (6.22)From (4.8) we note that U : ˚ H − / ( R ) (cid:55)→ ˚ H / ( R ) is an isometry. Then (6.22)and (4.19) imply that (cid:107) U S ( u + th ) − S ( u ) (cid:107) ˚ H / ( R ) → t → . (6.23)Using (6.14) we obtainlim t → t (cid:0) (cid:104) S ( u + th ) − S ( u ) , U S ( u ) (cid:105) + (cid:104) S ( u + th ) − S ( u ) , U S ( u + th ) − S ( u ) (cid:105) (cid:1) = 2 (cid:90) R S (cid:48) ( u ( x )) h ( x ) U S ( u ) ( x ) d x. (6.24)Hence, the assertion follows via (6.19). (cid:117)(cid:116) Proposition 6.2.
Let V ∈ ˚ H / ( R ) . Then E at every u ∈ H admits a direc-tional derivative with respect to test functions h ∈ C ∞ c ( R ) . Furthermore, thederivative is given by ddt E ( u + th ) (cid:12)(cid:12)(cid:12) t =0 = 2 a (cid:104) u, h (cid:105) ˚ H / ( R ) + (cid:90) R Φ (cid:48) ( u ( x )) h ( x ) d x + (cid:90) R V ( x ) S (cid:48) ( u ( x )) h ( x ) d x + b (cid:90) R U S ( u ) ( x ) S (cid:48) ( u ( x )) h ( x ) d x. (6.25) Proof.
Follows from Lemmas 6.1–6.3 and (5.4). (cid:117)(cid:116)
Remark 6.1.
The corresponding Euler–Lagrange equation is then in the distri-butional sense 0 = 2 a ( − ∆ ) / u + Φ (cid:48) ( u ) + V S (cid:48) ( u ) + bU S ( u ) S (cid:48) ( u ) . (6.26)Observing that Φ (cid:48) ( u ) = uS (cid:48) ( u ) and S (cid:48) ( u ) = 2 | u + ¯ u | , we rewrite (6.26) in theform 0 = a ( − ∆ ) / u + | u + ¯ u | (cid:0) u + V + bU S ( u ) (cid:1) . (6.27) Using the Euler–Lagrange equation for E we shall establish ad-ditional regularity of the minimizers. Lemma 6.4.
Assume that V ∈ ˚ H / ( R ) . Let u ∈ H be such that E ( u ) =inf ˜ u ∈H E (˜ u ) . Then u ∈ C / ( R ) ∩ L ∞ ( R ) and u ( x ) → as | x | → ∞ .Proof. Since u ∈ H is a minimizer of E , it satisfies the Euler-Lagrange equation(6.27) distributionally. Denote F ( x ) := − a − | u ( x ) + ¯ u | (cid:0) u ( x ) + V ( x ) + bU S ( u ) ( x ) (cid:1) , x ∈ R . (6.28)If F ∈ L s ( R ) for some 1 < s < u ( x ) = 12 π (cid:90) R F ( y ) | x − y | d y ∈ L t ( R ) , t = 1 s − , (6.29)see Lemma 4.1. So we can apply the bootstrap argument in an attempt toimprove the L t –regularity of u .First, we consider the case ¯ u = 0. Then u ∈ L p ( R ) for all p ∈ [3 , V, U S ( u ) ∈ L ( R ), we conclude that u ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:3) , (6.30) uV, uU S ( u ) ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:3) . (6.31)Then F ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:3) , (6.32)and therefore, by (6.29) and (5.12) u ∈ L t ( R ) , ∀ t ≥ . (6.33)Iterating once more, we deduce that F ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:1) . (6.34)Then by Lemma 4.1 and Remark 4.1 we obtain u ∈ C − s ( R ) ∩ L t ( R ) , ∀ t ∈ [3 , ∞ ] , ∀ s ∈ (cid:0) , (cid:1) . (6.35)In particular, this means that in (6.34) we can take s = 4. Applying Lemma 4.1once again with s = 4, we finally deduce that u ∈ C / ( R ) ∩ L t ( R ) , ∀ t ∈ [3 , ∞ ] . (6.36)Next consider the case ¯ u (cid:54) = 0. Then u ∈ L p ( R ) for all p ∈ [2 , V, U S ( u ) ∈ L ( R ), we conclude that u ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:3) , (6.37) uV, uU S ( u ) ∈ L s ( R ) ∀ s ∈ (cid:2) , (cid:3) , (6.38)¯ uV, ¯ uU S ( u ) ∈ L ( R ) . (6.39) rbital-free density functional theory of out-of-plane charge screening in graphene 29 Hence F = F + F , F ∈ L ( R ) , F ∈ L ( R ) , (6.40)and we do not gain at this point any additional regularity because of the lackof decay at infinity coming from ¯ uV and ¯ uU S ( u ) . Since the Riesz potential in(6.29) could be applied (as an integral operator) only to functions in L s ( R )with s <
2, the previous bootstrap procedure fails on the whole of R . Instead,we will use a localized version based on Lemma 4.2.Given arbitrary R >
0, we represent u = u R + h R , u R := u R, + u R, , (6.41)where u R, ( x ) := 12 π (cid:90) B R (0) F ( y ) | x − y | d y, u R, ( x ) := 12 π (cid:90) B R (0) F ( y ) | x − y | d y. (6.42)Since χ B R (0) F ∈ L s ( R ) for any s ∈ [1 , (cid:107) u R, (cid:107) L ∞ ( B R (0)) ≤ C R (cid:107) F (cid:107) L ( R ) , (6.43)for some C R > R (here and in the rest of the proof wesuppress the dependence of all the constants on a , b and ¯ ρ ). Similarly, since χ B R (0) F ∈ L s ( R ) for any s ∈ [1 , u R, ∈ L t ( R ) forall t >
2. Furthermore, by H¨older inequality we obtain (cid:107) u R, (cid:107) L t ( B R (0)) ≤ C R,t (cid:107) F (cid:107) L ( R ) , (6.44)for some C R,t > R and t . At the same time, the function h R := u − u R solves (cid:104) h R , ϕ (cid:105) ˚ H / ( R ) = (cid:90) R \ B R (0) F ( x ) ϕ ( x ) d x ∀ ϕ ∈ C ∞ c ( R ) . (6.45)Therefore, by Lemma 4.2 we have h R ∈ W , ∞ ( B R (0)), with (cid:107) h R (cid:107) L ∞ ( B R (0)) ≤ C R (cid:107) u (cid:107) L ( R ) for some C R > R . Thus, we have u ∈ L t ( B R (0))for any t >
2, with the norm controlled by constant depending only on R , t , (cid:107) u (cid:107) L ( R ) , (cid:107) V (cid:107) L ( R ) and (cid:107) U S ( u ) (cid:107) L ( R ) . Furthermore, by possibly increasing thevalue of the constant, we can make the same conclusion about (cid:107) u (cid:107) L t ( B R (0)) .Bootstrapping this information, we then obtain that χ B R (0) F ∈ L s ( R ) withany s ∈ [1 , u R, ∈ L ∞ ( B R (0)),with the norm controlled by (cid:107) u (cid:107) L ( R ) , (cid:107) V (cid:107) L ( R ) and (cid:107) U S ( u ) (cid:107) L ( R ) , and theconstant depending only on R . Combining this with the L ∞ -bounds on u R, and h R , we then conclude that (cid:107) u (cid:107) L ∞ ( B R (0)) ≤ C R for some constant C R > R and (cid:107) u (cid:107) L ( R ) , (cid:107) V (cid:107) L ( R ) and (cid:107) U S ( u ) (cid:107) L ( R ) . Furthermore,since the obtained estimates for fixed R > u ∈ L ∞ ( R ). The fact that u ∈ L ∞ ( R ) implies that F ∈ L ( R ). Noting that χ B R (0) F ∈ L s ( R ) for any s ∈ [1 , | u R ( x ) − u R ( y ) | ≤ C (cid:107) F (cid:107) L ( R ) | x − y | / ∀ x, y ∈ R , (6.46)for some universal C >
0. On the other hand, since (cid:107) h R (cid:107) W , ∞ ( B R (0)) → R → ∞ , fixing x and y and passing to the limit we conclude that | u ( x ) − u ( y ) | ≤ C (cid:107) F (cid:107) L ( R ) | x − y | / ∀ x, y ∈ R , (6.47)and, hence, u ∈ C / ( R ) ∩ L t ( R ) ∀ t ∈ [2 , ∞ ] . (6.48)Finally, it is standard to see that u ∈ C α ( R ) ∩ L p ( R ) for some α ∈ (0 ,
1] andsome p ≥ u ( x ) → | x | → ∞ . (cid:117)(cid:116) Remark 6.2.
The regularity of minimizers of E can be improved under addi-tional smoothness assumptions on V . For instance, assume that V ∈ ˚ H / ( R ) ∩ C / ( R ). Taking into account that S ( · ) is a C –mapping and using Lemma 4.2,similarly to the arguments in the proof of Lemma 6.4 one can show that U S ( u ) ∈ C / ( R ) ∩ L ∞ ( R ) as well. Then the expression | u ( x )+¯ u | (cid:0) u ( x ) + V ( x ) + bU S ( u ) ( x ) (cid:1) in the right hand side of (6.28) is a bounded, C / –H¨older continuous function,and we can conclude that u ∈ C , / ( R ) by [51, Proposition 2.8]. Furthermore,if we assume that V ∈ ˚ H / ( R ) ∩ C α ( R ) for some α ∈ ( , u ∈ C ,α ( R ).Note, however, that if u ∈ C ,α ( R ), but u + ¯ u changes sign then | u + ¯ u | , and,hence, the whole right hand side of (6.28), is merely a locally Lipschitz functionof x regardless of the smoothness of V . Thus, generally speaking, local regularityof u can not be improved beyond C ,α ( R ). Let u ∈ H be such that E ( u ) = inf ˜ u ∈H E (˜ u ). Clearly, E ( u ) ≤
0. In particular, a (cid:107) u (cid:107) H / ( R ) + (cid:104) V, U S ( u ) (cid:105) ˚ H / ( R ) + b (cid:107) U S ( u ) (cid:107) H / ( R ) ≤ . (6.49)Applying Cauchy-Schwarz inequality and then the fractional Sobolev inequality(4.4), we conclude that12 b (cid:107) V (cid:107) H / ( R ) ≥ a (cid:107) u (cid:107) H / ( R ) ≥ a √ π (cid:107) u (cid:107) L ( R ) . (6.50)Similarly, by (6.49) and Cauchy-Schwarz inequality we have2 (cid:107) V (cid:107) ˚ H / ( R ) ≥ b (cid:107) U S ( u ) (cid:107) ˚ H / ( R ) ≥ π / b (cid:107) U S ( u ) (cid:107) L ( R ) . (6.51)Next assume that the inequality opposite to the one in the statement ofthe theorem holds, namely that (cid:107) u (cid:107) L ∞ ( R ) ≥ ¯ u . Choose x ∗ ∈ R such that rbital-free density functional theory of out-of-plane charge screening in graphene 31 | u ( x ∗ ) | ≥ (cid:107) u (cid:107) L ∞ ( R ) . Then | u + ¯ u | ≤ (cid:107) u (cid:107) ∞ . Using the same notations as inthe proof of Lemma 6.4, by (6.28), (6.50), (6.51) and (4.4) we have (cid:107) F (cid:107) L ( R ) ≤ C (cid:107) u (cid:107) L ∞ ( R ) (cid:107) V (cid:107) ˚ H / ( R ) , (6.52)for some C > a and b . Therefore by (6.47) for any R > B R ( x ∗ ) u ≤ C (cid:107) u (cid:107) L ∞ ( R ) (cid:107) V (cid:107) ˚ H / ( R ) R / , (6.53)again, for some C > a and b .Now, set R = c (cid:107) V (cid:107) H / ( R ) , (6.54)where c > a and b chosen in such a way thatosc B R ( x ∗ ) u ≤ (cid:107) u (cid:107) L ∞ ( R ) . Then (cid:107) u (cid:107) L ( R ) ≥ (cid:90) B R ( x ∗ ) u d x ≥ πR (cid:107) u (cid:107) L ∞ ( R ) ≥ C (cid:107) u (cid:107) L ∞ ( R ) (cid:107) V (cid:107) − H / ( R ) , (6.55)for some C > a and b , which yields (cid:107) u (cid:107) L ( R ) (cid:107) V (cid:107) ˚ H / ( R ) ≥ C (cid:107) u (cid:107) L ∞ ( R ) ≥ C ¯ u. (6.56)In view of (6.50), we then conclude that (cid:107) V (cid:107) ˚ H / ( R ) ≥ C, (6.57)for some C > a , b and ¯ ρ , which completes the proof. (cid:117)(cid:116)
7. Proof of Theorems 3.2, 3.4 and 3.5
We introduce the function class H + := (cid:110) u ∈ H : u ≥ − ¯ u (cid:111) , (7.1)which is an equivalent way of writing the class A +¯ ρ . To study the variationalproblem for E on H + , let us define another energy functional E + , given by(6.1) in which the functions Φ ( u ) and S ( u ) are replaced by Φ + ( u ) and S + ( u ),respectively. The latter are obtained from the former by a reflecion around u = − ¯ u from the range u ≥ − ¯ u to u ≤ − ¯ u (see Fig. 3 and compare with Fig. 2): S + ( u ) := S ( | u + ¯ u | − ¯ u ) = 2¯ uu + u , (7.2) Φ + ( u ) := Φ ( | u + ¯ u | − ¯ u ) = 23 ( | u + ¯ u | − ¯ u ) − ¯ uS + ( u ) . (7.3)We also introduce the function class˜ H := (cid:110) u ∈ ˚ H / ( R ) : S + ( u ) ∈ ˚ H − / ( R ) } , (7.4) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) u (cid:70) (cid:43) a (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) u S (cid:43) b (cid:76) Fig. 3. (a) Plot of Φ + ( u ) and (b) plot S + ( u ) for ¯ u = 1. which is the analog of A ¯ ρ in the context of E + ( u ). Thus, the energy functional E + ( u ), defined for all u ∈ ˜ H , is given by E + ( u ) := a (cid:107) u (cid:107) H / ( R ) + (cid:90) R Φ + ( u ( x )) d x + (cid:104) V, U S + ( u ) (cid:105) ˚ H / ( R ) + b (cid:107) U S + ( u ) (cid:107) H / ( R ) . (7.5)Note that, by construction, if u ∈ H + then u ∈ ˜ H and E ( u ) = E + ( u ).Analogous to Proposition 6.1, we have Proposition 7.1. If V ∈ ˚ H / ( R ) , then there exists u ∈ H + such that E + ( u ) =inf u ∈ ˜ H E + ( u ) . Furthermore, E ( u ) = inf u ∈H + E ( u ) .Proof. Observe first that for any u ∈ ˜ H , we have | u + ¯ u | − ¯ u ∈ H + ⊂ ˜ H , and by(4.1), we have (cid:13)(cid:13) | u + ¯ u | − ¯ u (cid:13)(cid:13) H / ( R ) = 14 π (cid:90) (cid:90) R × R (cid:12)(cid:12) | u ( x ) + ¯ u | − | u ( y ) + ¯ u | (cid:12)(cid:12) | x − y | d x d y ≤ π (cid:90) (cid:90) R × R | u ( x ) − u ( y ) | | x − y | d x d y = (cid:107) u (cid:107) H / ( R ) . (7.6)Hence, E + ( | u + ¯ u | − ¯ u ) ≤ E + ( u ) . (7.7)Therefore, for a minimizing sequence { u n } of E + in ˜ H , we can consider { ˜ u n } := {| u n + ¯ u | − ¯ u } ⊂ H + , which is also a minimizing sequence. The existence of aminimizer then follows from the proof of Proposition 6.1 by changing S, Φ to S + , Φ + in that proof.Finally, E + ( u ) = E ( u ) for u ∈ H + , since S + ( u ), Φ + ( u ) coincide with S ( u ), Φ ( u ) for u ≥ − ¯ u . Therefore, the minimizer u of E + (taken to be in H + ) alsominimizes E over H + . (cid:117)(cid:116) It is also clear that any minimizer of E over H + is also a minimizer of E + over ˜ H . The advantage of considering E + is to remove the constraint u ≥ − ¯ u rbital-free density functional theory of out-of-plane charge screening in graphene 33 in H + . In particular, we can derive the Euler–Lagrange equation of E + for aminimizer u ∈ H + , observing that the arguments in Section 6.2 apply verbatimto the functional E + (by replacing S and Φ with S + and Φ + , respectively). If u ∈ H + is a minimizer of E + , it then satisfies the Euler-Lagrange equation givenin the distributional sense by0 = a ( − ∆ ) / u + | u + ¯ u | ( u + V + bU S + ( u ) ) . (7.8)Note that since u ≥ − ¯ u , the absolute value can be omitted and S + ( u ) coincideswith S ( u ) in the above equation. Using the Euler–Lagrange equation, we shallestablish additional properties for the minimizer. Lemma 7.1.
Assume V ∈ ˚ H / ( R ) . Let u ∈ H + be such that E ( u ) = inf ˜ u ∈H + E (˜ u ) .Then u ∈ C / ( R ) ∩ L ∞ ( R ) , u ( x ) → as | x | → ∞ and u ( x ) > − ¯ u for all x ∈ R .Proof. The regularity follows verbatim from the proof of Lemma 6.4. Also, since u ∈ L ∞ ( R ), we have S ( u ) ∈ L ( R ), and we can again repeat the argumentsin the proof of Lemma 6.4, now applied to (5.10), to establish that U S ( u ) ∈ C / ( R ) ∩ L ∞ ( R ) as well.Now, since u satisfies (7.8) and u ≥ − ¯ u , we have that w := u + ¯ u ≥ a ( − ∆ ) / w + V w + w ( u + bU S ( u ) ) . (7.9)Note that by the argument at the beginning of the proof we have w ∈ L ∞ ( R )and | u + bU S ( u ) | ≤ c, (7.10)for some c > x ∈ R . Decompose V = V + − V − , where V + and V − arethe positive and the negative part of V , respectively. Then a ( − ∆ ) / w + V + w + cw = V − w + ( c − ( u + bU S ( u ) )) w ≥ . (7.11)Since V + ∈ ˚ H / ( R ) ⊂ L ( R N ), the potential V + + c belongs to the local Katoclass K / loc with respect to ( − ∆ ) / , i.e., for every ball B ⊂ R we havelim ε → sup x ∈ R (cid:90) B ε ( x ) ( V + ( y ) + c ) χ B ( y ) | x − y | d y = 0 , (7.12)see [30, Definition 2.1] or [11, Section III and Theorem III.1(iii)]. Then standardmethods of semigroup theory (see e.g. [46, Section XIII.12]) can be used to showthat a ( − ∆ ) / + V + + c defines a self-adjoint positive-definite linear operator in L ( R ) [30, Theorem 2.1]. Moreover, the Green’s function G V + a,c ( x, y ) : R × R → R of a ( − ∆ ) / + V + + c is well defined and strictly positive (cf. [30, Lemma 2.1(4)and Section 2.4]). In addition, since V + ≥
0, the Green’s function G V + a,c ( x, y ) isdominated by the Green’s function G a,c ( | x − y | ) of the operator a ( − ∆ ) / + c ,so that we have0 < G V + a,c ( x, y ) ≤ G a,c ( | x − y | ) for all x, y ∈ R . (7.13) The function G a,c ( r ) is given explicitly by G a,c ( r ) := c a (cid:18) aπcr − HHH (cid:16) cra (cid:17) + Y (cid:16) cra (cid:17)(cid:19) , (7.14)where HHH ( z ) is the Struve function, Y ( z ) is the Bessel function of the secondkind, which can be obtained using Fourier transform. Moreover, G a,c obeys [22,Theorem 3.3 and Lemma 4.1] G a,c ( r ) ∼ (cid:40) r − , r (cid:28) ,r − , r (cid:29) , (7.15)and, therefore, we have G V + a,c ∈ L / ( R ) ∩ L ( R ). Denoting the right-hand sideof (7.11) by g ( x ) ≥
0, since g ∈ L ( R ) + L ∞ ( R ) we then have distributionallyand a.e. in R w ( x ) = (cid:90) R G V + a,c ( x, y ) g ( y ) d y. (7.16)This implies that w is strictly positive. (cid:117)(cid:116) Remark 7.1.
Note that unlike minimizers in H (see Remark 6.2), further reg-ularity of minimizers u ∈ H + is expected under additional smoothness hy-pothesis on V . For example, if we assume that V ∈ ˚ H / ( R ) ∩ C ∞ ( R ) then u ∈ C ∞ ( R ). Indeed, if u ∈ C k,α ( R ) for some k ≥ w := u + ¯ u > w (cid:0) u + V + bU S ( u ) (cid:1) ∈ C k,α ( R ). Thus, differentiating the Euler–Lagrange equa-tion in (7.9) with respect to x and applying [51, Proposition 2.8], we concludethat u ∈ C k +1 ,α ( R ). This argument can be iterated infinitely many times, weomit the details.Finally, we show that the minimizer of E on H + is unique. It is more conve-nient to rewrite the energy functional using ρ as the variable, as in the uniquenessproof for the usual Thomas-Fermi-von Weizs¨acker model [35]. We write E ( ρ ) = a (cid:107)√ ρ − √ ¯ ρ (cid:107) H / ( R ) + (cid:90) R Φ (cid:0)(cid:112) ρ ( x ) − √ ¯ ρ (cid:1) d x + (cid:104) V, U S ( √ ρ −√ ¯ ρ ) (cid:105) ˚ H / ( R ) + b (cid:107) U S ( √ ρ −√ ¯ ρ ) (cid:107) H / ( R ) . (7.17)Note that S ( √ ρ − √ ¯ ρ ) = ρ − ¯ ρ, (7.18)and is, therefore, linear in ρ , and Φ ( √ ρ − √ ¯ ρ ) = 23 ( ρ / − ¯ ρ / ) − √ ¯ ρ ( ρ − ¯ ρ ) (7.19)is strictly convex in ρ . Hence, the last three terms in E ( ρ ) given by (7.17) areconvex on A +¯ ρ . Moreover, even though √ ρ is a concave function of ρ , the followinglemma shows that (cid:107)√ ρ − √ ¯ ρ (cid:107) H / ( R ) is convex, and the energy E ( ρ ) is strictlyconvex. The uniqueness of the minimizer then follows. rbital-free density functional theory of out-of-plane charge screening in graphene 35 Lemma 7.2.
The set A +¯ ρ is convex. Furthermore, the functional E ( ρ ) defined in (7.17) is strictly convex on A +¯ ρ , i.e., for every ρ , ρ ∈ A +¯ ρ , ρ (cid:54) = ρ , and every t ∈ (0 , , there holds E ( tρ + (1 − t ) ρ ) < tE ( ρ ) + (1 − t ) E ( ρ ) . (7.20) Proof.
Denote ρ t = tρ + (1 − t ) ρ . From [37, Theorem 7.13] it follows that √ ρ t − √ ¯ ρ ∈ ˚ H / ( R ) and (cid:107)√ ρ t − √ ¯ ρ (cid:107) H / ( R ) ≤ t (cid:107)√ ρ − √ ¯ ρ (cid:107) H / ( R ) + (1 − t ) (cid:107)√ ρ − √ ¯ ρ (cid:107) H / ( R ) . (7.21)Also, clearly ρ t − ¯ ρ ∈ ˚ H − / ( R ) and ρ t ≥
0. Hence, ρ t ∈ A +¯ ρ , implying that A +¯ ρ is a convex set. The strict convexity of E ( ρ ) then follows from the strictlyconvexity of Φ in the second term in E ( ρ ). (cid:117)(cid:116) For u ∈ H + , we have E ( u ) = E + ( u ), where E + isdefined in (7.5) with the specific choice ¯ u = 0: E + ( u ) = a (cid:107) u (cid:107) H / ( R ) + 23 (cid:90) R | u ( x ) | d x − (cid:90) R | u ( x ) | (cid:0) | x | (cid:1) / d x + b (cid:107) U | u | (cid:107) H / ( R ) . (7.22)In view of Theorem 3.4, in order to prove Theorem 3.5 it is sufficient to showthat:( i ) If a ≥ a c , then E ( u ) > u ∈ H ,( ii ) If a < a c , then inf u ∈H + E + ( u ) < i ) follows directly from the fractional Hardy’s inequality a c (cid:107) u (cid:107) H / ( R ) ≥ (cid:90) R | u ( x ) | | x | d x, (7.23)which is valid for all u ∈ ˚ H / ( R ) with the optimal constant a c = Γ (1 / Γ (3 / , see[24, Remark 4.2].Claim ( ii ) is a consequence of the following. Lemma 7.3.
Let c < a c . Then there exists u c ∈ C ∞ c ( R ) such that u c ≥ and c (cid:107) u c (cid:107) H / ( R ) < (cid:90) R | u c ( x ) | (cid:0) | x | (cid:1) / d x. (7.24)Indeed, let a < a c . Then, using Lemma 7.3 with some c ∈ ( a, a c ), for allsufficiently small t > E + ( tu c ) < − ( c − a ) t (cid:107) u c (cid:107) H / ( R ) + 2 t (cid:90) R | u c ( x ) | d x + bt (cid:107) U | u c | (cid:107) H / ( R ) < . (7.25) We conclude that inf u ∈H + E + ( u ) <
0, which proves Claim ( ii ).We are only left to prove Lemma 7.3. Proof (of Lemma 7.3).
Let u ∈ C ∞ c ( R ) be such that c (cid:107) u (cid:107) H / ( R ) − (cid:90) R | u ( x ) | | x | d x ≤ − u ∈ C ∞ c ( R ) as a suitable approxi-mation of | x | − / ). For λ >
0, set u λ ( x ) = u ( x/λ ). Then c (cid:107) u λ (cid:107) H / ( R ) − (cid:90) R | u λ ( x ) | (cid:0) | x | (cid:1) / d x = λ (cid:32) c (cid:107) u (cid:107) H / ( R ) − (cid:90) R | u ( y ) | (cid:0) λ − + | y | (cid:1) / d y (cid:33) ≤ − λ , (7.27)for all sufficiently large λ >
0, in view of (7.26) and the monotonicity of themapping λ (cid:55)→ ( λ − + | y | ) − / . (cid:117)(cid:116) Acknowledgments
The authors wish to thank an anonymous referee for helpful suggestions. JLwould like to acknowledge support from the Alfred P. Sloan Foundation and theNational Science Foundation under award DMS-1312659. CBM was supported,in part, by the National Science Foundation via grants DMS-0908279 and DMS-1313687.
References [1] D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty,
Propertiesof graphene: a theoretical perspective , Adv. Phys. (2010), 261–482.[2] D. S. L. Abergel, P. Pietil¨ainen, and T. Chakraborty, Electronic compressibility ofgraphene: The case of vanishing electron correlations and the role of chirality , Phys.Rev. B (2009), 081408.[3] T. Ando, Screening effect and impurity scattering in monolayer graphene , J. Phys. Soc.Jap. (2006), 074716.[4] D. H. Armitage, A counter-example in potential theory , J. London Math. Soc. (2) (1975), 16–18.[5] Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A. H. MacDonald, Chirality andcorrelations in graphene , Phys. Rev. Lett. (2007), 236601.[6] R. D. Benguria, H. Brezis, and E. H. Lieb, The Thomas-Fermi-von Weizs¨acker theory ofatoms and molecules , Commun. Math. Phys. (1981), 167–180.[7] R. D. Benguria, M. Loss, and H. Siedentop, Stability of atoms and molecules in an ultra-relativistic Thomas-Fermi-Weizs¨acker model , J. Math. Phys. (2008), 012302.[8] L. Brey and H. A. Fertig, Linear response and the Thomas-Fermi approximation inundoped graphene , Phys. Rev. B (2009), 035406.[9] H. Br´ezis and F. Browder, A property of Sobolev spaces , Comm. Partial Differential Equa-tions (1979), 1077–1083.rbital-free density functional theory of out-of-plane charge screening in graphene 37[10] E. Canc`es and V. Ehrlacher, Local defects are always neutral in the Thomas–Fermi–vonWeisz¨acker theory of crystals , Arch. Ration. Mech. Anal. (2011), 933–973.[11] R. Carmona, W. C. Masters, and B. Simon,
Relativistic Schr¨odinger operators: asymptoticbehavior of the eigenfunctions , J. Funct. Anal. (1990), 117–142.[12] H. Cartan, Th´eorie du potentiel newtonien: ´energie, capacit´e, suites de potentiels , Bull.Soc. Math. France (1945), 74–106.[13] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Theelectronic properties of graphene , Rev. Mod. Phys. (2009), 109–162.[14] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene , Rev. Mod. Phys. (2011), 407–470.[15] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolevspaces , Bull. Sci. Math. (2012), 521–573.[16] D. P. DiVincenzo and E. J. Mele,
Self-consistent effective-mass theory for intralayerscreening in graphite intercalation compounds , Phys. Rev. B (1984), 1685–1694.[17] N. du Plessis, An introduction to potential theory , Hafner Publishing Co., Darien, Conn.,1970. University Mathematical Monographs, No. 7.[18] E. Engel and R. M. Dreizler,
Field-theoretical approach to a relativistic Thomas-Fermi-Weizs¨acker model , Phys. Rev. A. (1987), 3607–3618.[19] , Solution of the relativistic Thomas-Fermi-Dirac-Weizs¨acker model for the caseof neutral atoms and positive ions , Phys. Rev. A. (1988), 3909–3917.[20] C. L. Fefferman and M. I. Weinstein, Honeycomb lattice potentials and Dirac points , J.Amer. Math. Soc. (2012), 1169–1220.[21] , Wave packets in honeycomb structures and two-dimensional Dirac equations ,Comm. Math. Phys. (2014), 251–286.[22] P. Felmer, A. Quaas, and J. Tan,
Positive solutions of the nonlinear Schr¨odinger equationwith the fractional Laplacian , Proc. Roy. Soc. Edinburgh Sect. A (2012), 1237–1262.[23] M. M. Fogler, D. S. Novikov, and B. I. Shklovskii,
Screening of a hypercritical charge ingraphene , Phys. Rev. B (2007), 233402.[24] R. L. Frank, E. H. Lieb, and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractionalSchr¨odinger operators , J. Amer. Math. Soc. (2008), 925–950.[25] J. Garc´ıa-Cuerva and A. E. Gatto, Boundedness properties of fractional integral operatorsassociated to non-doubling measures , Studia Math. (2004), 245–261.[26] A. K. Geim and K. S. Novoselov,
The rise of graphene , Nat. Mater. (2007), 183–191.[27] J. Gonz´alez, F. Guinea, and M. A. H. Vozmediano, Non-fermi liquid behavior of electronsin the half-filled honeycomb lattice (a renormalization group approach) , Nucl. Phys. B (1994), 595–618.[28] C. Hainzl, M. Lewin, and C. Sparber,
Ground state properties of graphene in Hartree-Focktheory , J. Math. Phys. (2012), 095220.[29] E. H. Hwang and S. Das Sarma, Dielectric function, screening, and plasmons in two-dimensional graphene , Phys. Rev. B (2007), 205418.[30] K. Kaleta and J. L˝orinczi, Fractional P ( φ ) -processes and Gibbs measures , StochasticProcess. Appl. (2012), 3580–3617.[31] M. I. Katsnelson, Nonlinear screening of charge impurities in graphene , Phys. Rev. B (2006), 201401(R).[32] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H. Castro Neto, Electron-electroninteractions in graphene: Current status and perspectives , Rev. Mod. Phys. (2012),1067–1125.[33] N. S. Landkof, Foundations of modern potential theory , Springer-Verlag, New York, 1972.Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischenWissenschaften, Band 180.[34] C. Le Bris and P.-L. Lions,
From atoms to crystals: a mathematical journey , Bull. Amer.Math. Soc. (N.S.) (2005), 291–363.[35] E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules , Rev. Mod. Phys. (1981), 603–641.[36] E. H. Lieb and M. Loss, Analysis , Second, Graduate Studies in Mathematics, vol. 14,American Mathematical Society, Providence, RI, 2001.8 Jianfeng Lu, Vitaly Moroz, Cyrill B. Muratov[37] E. H. Lieb, M. Loss, and H. Siedentop,
Stability of relativistic matter via Thomas-Fermitheory , Helv. Phys. Acta (1996), 974–984.[38] E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids ,Advances in Math. (1977), 22–116.[39] E. H. Lieb and H.-T. Yau, The stability and instability of relativistic matter , Comm.Math. Phys. (1988), 177–213.[40] J. Lu, V. Moroz, and C. B. Muratov, 2014. In preparation.[41] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A.Yacoby,
Observation of electron-hole puddles in graphene using a scanning single-electrontransistor , Nat. Phys. (2008), 144–148.[42] V. G. Maz (cid:48) ja and V. P. Havin, A nonlinear potential theory , Uspehi Mat. Nauk (1972),67–138.[43] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V.Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films , Science (2004), 666–669.[44] M. Polini, A. Tomadin, R. Asgari, and A. H. MacDonald,
Density functional theory ofgraphene sheets , Phys. Rev. B (2008), 115426.[45] J. P. Reed, B. Uchoa, Y. I. Joe, Y. Gan, D. Casa, E. Fradkin, and P. Abbamonte, Theeffective fine-structure constant of freestanding graphene measured in graphite , Science (2010), 805–808.[46] M. Reed and B. Simon,
Methods of modern mathematical physics. IV. Analysis of oper-ators , Academic Press, New York, 1978.[47] S. Rempel, ¨Uber die Nichtvollst¨andigkeit eines Raumes von Ladungen mit endlicher En-ergie , Math. Nachr. (1976), 87–91.[48] D. Ruiz, On the Schr¨odinger-Poisson-Slater system: behavior of minimizers, radial andnonradial cases , Arch. Ration. Mech. Anal. (2010), 349–368.[49] K. W. K. Shung,
Dielectric function and plasmon structure of stage-1 intercalatedgraphite , Phys. Rev. B (1986), 979–993.[50] A. V. Shytov, M. I. Katsnelson, and L. S. Levitov, Vacuum polarization and screening ofsupercritical impurities in graphene , Phys. Rev. Lett. (2007), 236801.[51] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplaceoperator , Comm. Pure Appl. Math. (2007), 67–112.[52] I. Sodemann and M. M. Fogler, Interaction corrections to the polarization function ofgraphene , Phys. Rev. B (2012), 115408.[53] E. M. Stein, Singular integrals and differentiability properties of functions , PrincetonMathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.[54] M. Struwe,
Variational methods , Springer-Verlag, Berlin, 1990.[55] P. R. Wallace,
The band theory of graphite , Phys. Rev. (1947), 622–634.[56] J. Wang, H. A. Fertig, G. Murthy, and L. Brey, Excitonic effects in two-dimenaionalmassless Dirac fermions , Phys. Rev. B (2011), 035404.[57] Y. Wang, V. W. Brar, A. V. Shytov, Q. Wu, W. Regan, H.-Z. Tsai, A. Zettl, L. S. Levitov,and M. F. Crommie, Mapping Dirac quasiparticles near a single Coulomb impurity ongraphene , Nat. Phys. (2012), 653–657.[58] G. L. Yu, R. Jalil, B. Bell, A. S. Mayorov, P. Blake, F. Schedin, S. V. Morozov, L. A.Ponomarenko, F. Chiappini, S. Wiedmann, U. Zeitler, M. I. Katsnelson, A. K. Geim, K. S.Novoselov, and D. C. Elias, Interaction phenomena in graphene seen through quantumcapacitance , Proc. Natl. Acad. Sci. USA (2013), 3282–3286.[59] L. M. Zhang and M. M. Fogler,
Nonlinear screening and ballistic transport in a graphenep-n junction , Phys. Rev. Lett.100