NONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL. x y

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1 NONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL JIANFENG LU AND FELIX OTTO In this paper, we study the following energy functional () E(ϕ) := ϕ 2 + F(ϕ 2 ) dx + D(ϕ 2,ϕ 2 ), R 3 where F(t) = t 5/3 t 4/3 and D(, ) is the Coulomb interaction in R 3 : f(x)g(y) D(f,g) = dx dy. R 3 R 3 Our main result is the following nonexistence result. Theorem. There exists constant m 0 > 0, such that the variational problem (2) inf R ϕ 2 =m E(ϕ) does not have a minimizer if m m 0. We also consider a sharp interface version of the functional (). For three-dimensional sets R 3, define (3) E() := + dx dy. A nonexistence result similar to Theorem also holds. Theorem 2. There exists constant V 0 > 0, such that the variational problem (4) inf E() =V does not have a minimizer if V V 0. If the Coulomb term is not present in () and (3), a standard Modica-Mortela type result establishes a link between the two functional through Gamma convergence when the volume constraint m is large. This is no longer valid for the energy functionals () and (3) due to the Coulomb term. The connection between the two functionals is just on the formal level. Part of this work is done during J.L. s visits to the Max Planck Institute for Mathematics in the Sciences. He is grateful for the hospitality of the institute. We would like to thank Rustum Choksi, Robert V. Kohn, Cyrill Muratov, and Mark Peletier for helpful discussions.

2 We will prove Theorem and 2 separately. While our proof for Theorem can also be used to prove Theorem 2 after straightforward modifications, it is not clear how to extend our proof for Theorem 2 to the diffuse interface case (). The rest of the article is organized as follows. After discussion of motivations and some remarks, we first present the proof of Theorem 2 in Section. The proof of Theorem follows in Section 2. Physically, the functional () is known as the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) model for electrons with no external potential, where = ϕ 2 is the electron density. A mathematical introduction to the model and its physical motivation can be found in [2, 3, 5] and references therein. It is also related with the Schrödinger-Poisson-Slater system, see for example [9,20]. The sharp interface functional (3) recently receives attention from many authors, see for example [, 4, 6, 7,, 8, 25]. It is a natural extension of the isoperimetric problem. It is connected to various mean-field type models such as Ohta-Kawasaki, Ginzburg-Landau, and Thomas-Fermi type models. The version of Theorem 2 in two space dimensions has recently been proved in []. The nonexistence problem of (2) arises when we consider the small volume-fraction limit for electrons in a box with uniform background charge (physically known as the Jellium model). Using physical arguments, one expects the formation of crystalline structures made of separated electron bubbles, known as Wigner crystallization [26]. The proof of Wigner crystallization is an open challenging problem. If the TFDW model is used, the energy functional () then describes the self-energy of such a single electron bubble. Theorem states that such a bubble can only contain a finite amount of electrons, and hence, the system will indeed break up into many bubbles in the limit. We hope the result here can be used to understand Wigner crystallization in TFDW model, which is still open. Related to the Wigner crystallization, the small volume-fraction limit for an Ohta-Kawasaki type model was studied in [4,5], where they assume Theorem 2 as part of their conjecture in [5, Section 6] to make sure that multiple bubbles arise. See also small volume-fraction limits for related models studied recently in [0,7,2]. Physically, Theorem can also be interpreted as the vacuum can only bind a finite number of electrons in the TFDW model. This is related to the ionization conjecture in quantum mechanics, which states that the number of electrons that can be bound to an atomic nucleus of charge Z cannot exceed Z +. The ionization conjecture, while still open for the Schrödinger equation, has been studied by many authors for different types of models in quantum mechanics (see e.g. [2,3,8,4,6,22 24]). In particular, for the Thomas-Fermivon Weizsäcker model, where the non-convex term in the TFDW model ϕ 8/3 is dropped, the ionization conjecture was proved in [3]. For the TFDW model, however, it is still open whether an atom nucleus can only bind a finite number of electrons. Theorem confirms 2

3 this for Z = 0 (system without atoms). Our proof for Theorem however does not cover the case with external potential caused by an atomic nucleus, since we have used translational invariance in the proof. The case with atoms will be investigated in future work.. Isoperimetric problem with nonlocal interaction We prove Theorem 2 in this section. We start the proof with the following lemma collects two elementary properties of the variational problem (4). Lemma.. Let V. (i) We have E(V ) := inf{e() = V } V. (ii) Let be a minimizer of (4), is then connected. (Connectedness is defined in a measure-theoretic sense). Proof. We start by establishing the inequality (5) E(V ) E(V 0 ) + E(V ) for V = V 0 + V. To prove this inequality, we first argue that E(V ) could have also been defined on the basis of bounded sets: (6) E(V ) = inf{ E() = V, is bounded }. This inequality (5) can be seen as follows: For given with = V and E() < we have to argue that there exists a bounded with = V and such that E( ) is only slightly larger than E(). Indeed, because of <, there exists a possibly large R < such that \B R (0) and B R (0). Consider the stretched, cut-off set := λ( B R (0)), V where λ := is defined such that V \B R = V. We have for the Coulomb energy (0) and for the interfacial energy e e dx dy = λ5 λ 5 Hence we obtain for the total energy B R (0) B R (0) dx dy dx dy = λ 2 ( B R (0)) λ 2 ( + B R (0) ). E( ) λ 5 E() + λ 2 B R (0) and the last expression is E() since λ and B R (0). We now turn to inequality (5). According to (6), for given ǫ > 0, there exist bounded sets i, i = 0,, with E( i ) E(V i ) + ǫ. Let R < be so large that i B R (0). 3

4 Define := 0 + (d + ) with a shift vector d with d > 2R. Note that for x 0 and y d + we have d 2R > 0. In particular, we have = 0 + and = 0 + = V 0 + V, so that we may use in the definition of E(V ): E(V ) E( ) E( 0 ) + E( ) + V 0V d 2R E(V 0 ) + E(V ) + 2ǫ + V 0V d 2R. Letting first d tend to infinity and then ǫ to zero yields (5). Argument for (i). If is a ball of volume V, we have E() V 2/3 + V 5/3. In particular, if is a ball of volume V, we have E(). We reformulate this as E(V ) for V < 2. Using the subadditivity (5), we obtain by induction in N E(V ) N for N V < N +. Argument for (ii). Later in the proof, we will need connectedness only in the following measure-theoretic sense: Suppose that for some ball B R we have both B R and \B R in the following sense (7) B R > 0 and \B R > 0. Then we claim that we have B R in the sense of (8) B R > 0. (For a Caccioppoli set, its characteristic function χ admits an inner trace χ τ on B R. Thus (8) means B R χ τ dh 2 > 0.) We establish (8) by contradiction. Suppose it were not true. Let us introduce 0 := B R and := \B R. Since by assumption (7), V 0 := 0 > 0 and V := > 0 we have for the mixed Coulomb energy 0 0 dx dy > 0, so that we obtain for the total Coulomb energy dx dy > dx dy + 0 dx dy. For the interfacial energy we have because of our erroneous assumption B R = 0: =

5 Combining the two last inequalities, we obtain E(V ) = E() > E( 0 ) + E( ) E(V 0 ) + E(V ), which in view of the obvious V = V 0 + V is in contradiction to the subadditivity (5). The following crucial lemma amounts to an elementary regularity estimate for a minimizer. In the rough terms of volume, it states that the set cannot be thinner than order one. It is based on the global a priori estimate in Lemma.(i) and an elementary argument from the theory of minimal surfaces, see for instance [9, Proposition 5.4], that we slightly adapt to deal with the perturbation through the volume constraint and the Coulomb energy. These two effects are lower order perturbations in the sense that they only affect length scales of order one or larger, but do not affect the regularity on length scales small compared to one. Lemma.2. Let be a minimizer of (4) with prescribed volume V. Then for R and all x such that B r (x) > 0, r > 0, (9) B R (x) R 3. Proof. Due to translational invariance, we can assume without loss of generality that x = 0. It is obviously enough to show (9) for (0) R, in particular R 3 V. We will compare to given by () := λ(\br ), where the stretching factor λ > makes sure that = V. Hence λ is given by (2) λ 3 := V \B R = V V B R. We note that because of B R R 3 and (0) we have (3) λ 3 B R V. We now turn to a comparison of the energies: For the Coulomb energy we have () dx dy = λ 5 dx dy e e B R B R λ 5 dx dy. 5

6 For the interfacial energy we note = λ 2 (\B R ) Hence from E() E( ) we obtain () = λ 2 ( + B R B R ) λ λ 5 + λ 2 B R B R. B R (λ 5 )E() + λ 2 B R, which using ( B R ) = B R + B R we rewrite as (4) ( B R ) (λ 5 )E() + (λ 2 + ) B R. Now convert the inequality (4) into a suitable estimate. We use the estimates the isoperimetric inequality B R 2/3 ( B R ), the global energy bound E() V, cf. Lemma.(i), the estimate λ 5 (3) (2) = B R, λ 3 V the estimate λ 2 in particular following from (3). We so obtain B R 2/3 B R + B R. Since B R R 3 (0), we have B R B R 2/3 so that the last estimate simplifies to B R 2/3 B R. We finally note that B R = d dr B R (and that B R > 0 by assumption). Hence the last estimate turns into the differential inequality d dr B R /3. Integrating this inequality yields the desired lower bound. Together with Lemma.(i), the next lemma yields Theorem 2 as a corollary. It combines Lemma.(ii) with Lemma.2 in an elementary way: Lemma.3. Let be a minimizer of prescribed volume V. Then we have for the Coulomb energy (5) dx dy V ln V. 6

7 Proof. We will first argue that Lemma.(ii) and Lemma.2 imply that cannot be too thin in the sense of (6) B R (x) R for all points x and radii < R 2 diam(). Indeed, fix a point x and a radius with < R diam(). Because of the latter, 2 is not contained in B R (x). According to the connectedness of as in Lemma.(ii), there exist points x,x 2,,x R such that (7) x i B R i for i =, 2,, R. (Indeed, we have for the inner trace χ i of the characteristic function χ of that B R i χ i dh 2 > 0. Take an dh 2 BR i -Lebesgue point x i of χ i with χ i (x i ) =. It will satisfy x i B R i and B r (x i ) > 0 for any r > 0.) Here R is the largest integer less than R. It is clear from (7) that the balls {B /2 (x i )} i=,, R are pairwise disjoint. According to Lemma.2 we have This yields (6). B R (x) R B /2 (x i ) i= (9) R. We now argue that (6) implies dx dy V ln diam(). The desired estimate (5) then follows from this estimate combined with diam V /3. We first note that for any R diam() we have for the six-dimensional volume 2 { (x,y) < R } = { y < R } dx Hence we have as desired dx dy = = = 0 0 = B R (x) dx (6) R. { (x,y) < R } dr R d { (x,y) < R } dr R dr { (x,y) < R } dr R2 0 2 diam() dr. R 7

8 2. Thomas-Fermi-Dirac-von Weizsäcker model We prove Theorem in this section. Assume the minimizer of (2) exists, denote it by ϕ. We first note an elementary property of ϕ: Lemma 2.. We have ϕ(x) [0, (4/5) 3/2 ] and ϕ 2 (x) F(ϕ 2 (x)) 0 for a.e. x R 3. Proof. Since E( ϕ ) E(ϕ), we may restrict to ϕ 0. Note that (4/5) 3 = argmin t 0 F(t). Take ϕ = ϕ (4/5) 3/2 and let m = ϕ 2 m. Since ϕ minimizes (2), we have (8) E(ϕ) E(ϕ ) + inf R ψ 2 =m m E(ψ). Indeed, for any ψ with ψ 2 = m m, let ψ L (x) = Z L [ϕ (x) + ψ(x + Le )], where e = (, 0, 0) R 3 and Z L is a normalization constant so that ψ 2 L = ϕ 2 = m. Taking L, it is clear that Z L and lim sup E(ψ L ) = E(ϕ ) + E(ψ). L We arrive at (8) as E(ϕ) E(ψ L ) for any L. Consider the scaling ψ η = η 3/2 ψ(x/η), so ψ 2 η = ψ 2. It is not hard to see that E(ψ η ) 0 as η, as each term in () goes to zero. Therefore, we have E(ϕ) E(ϕ ) as R inf E(ψ) 0. ψ 2 =m m Suppose {ϕ > (4/5) 3/2 } > 0, then as (4/5) 3 is the unique minimizer of F(t) for t 0, F(ϕ 2 ) < F(ϕ 2 ). Note that ϕ ϕ and D(ϕ 2,ϕ 2 ) D(ϕ 2,ϕ 2 ), we arrive at a contradiction E(ϕ ) < E(ϕ). Therefore, ϕ (4/5) 3/2. The remainder of the lemma follows easily. The next lemma establishes a key estimate of the minimizer. Lemma 2.2. For any x R 3 and two radii r,r > 0 such that R > r +, we have (9) ϕ 2 (y) dy ϕ 2 (y) dy ϕ 2 (y) dy. R + r B r+ (x)\b r(x) B r(x) B R (x)\b r+ (x) Proof. By translational invariance, it suffices to prove the lemma for x = 0. To prove the lemma, we will compare the energy of ϕ with a competitor obtained by cutting and moving part of the minimizer to infinity. Let us take two functions f,f 2 : R [0, ] satisfying the following properties: (i) f (t) =,t 0 and f (t) = 0,t ; 8

9 (ii) f C (R); (iii) f 2 + f 2 2 = ; (iv) f (t), f 2(t) 2. Define cutoff functions χ,χ 2 : R 3 [0, ] as χ (x) = f ( x r) and χ 2 (x) = f 2 ( x r). Let ϕ = χ ϕ and ϕ 2 = χ 2 ϕ. Use a similar argument leading to (8), we have (20) E(ϕ) E(ϕ ) + E(ϕ 2 ). Let us compare the three terms in the definition of the energy (). We start with ϕ 2 + ϕ 2 2 ϕ 2 = (χ ϕ) 2 + (χ 2 ϕ) 2 ϕ 2 = ϕ 2 ( χ 2 + χ 2 2 ) + 2(χ r χ ϕ r ϕ + χ 2 r χ 2 ϕ r ϕ). where r is the derivative in the radial direction. Note that Therefore, χ r χ + χ 2 r χ 2 = 2 r(χ 2 + χ 2 2) = 0. ϕ 2 + ϕ 2 2 ϕ 2 = ϕ 2 ( χ 2 + χ 2 2 ) ϕ 2 Br+ \B r. Integrating the last inequality, we obtain (2) ϕ 2 + ϕ 2 2 ϕ 2 ϕ 2. R 3 B r+ \B r For the nonlinear term, using Lemma 2., we estimate F(ϕ 2 ) + F(ϕ 2 2) F(ϕ 2 ) F(ϕ 2 ) Br+ \B r ϕ 2 Br+ \B r. Therefore, (22) F(ϕ 2 ) + F(ϕ 2 2) F(ϕ 2 ) ϕ 2. R 3 B r+ \B r For the Coulomb interaction, D(ϕ 2,ϕ 2 ) + D(ϕ 2 2,ϕ 2 2) D(ϕ 2,ϕ 2 ϕ 2 ) = 2 (x)ϕ 2 2(y) dx dy R 3 R 3 ϕ 2 2 (x)ϕ 2 2(y) dx dy B r B R \B r+ (23) ϕ 2 (x)ϕ 2 (y) = 2 dx dy B r B R \B r+ 2 ϕ 2 (x)ϕ 2 (y) dx dy, r + R B R \B r+ where we have used the fact that for x B r and y B R \B r+, r + R. The lemma follows by combining (20), (2), (22), and (23). 9 B r

10 Lemma 2.3. There exists a universal constant C 0, such that for any ball B R0 (x) with R 0 > and ϕ 2 C 0, B R0 (x) it holds (24) m = ϕ 2 2 R 3 B 2R0 (x) Proof. Without loss of generality, we assume x = 0 to simplify notations. Consider balls B r concentric with B R0. Denote for r 0 V (r) = ϕ 2. B r ϕ 2. From Lemma 2.2, we have for any R > and r (0,R ), V (r)(v (R) V (r + )) (25) V (r + ) V (r). R + r Hence, for R > 4 and r [R/4,R/2 ] (so that r < R ), by monotonicity of V (r), V (r)(v (R) V (r + )) (26) V (r + ) V (r) R Integrating r from R/4 to R/2, we obtain V (R/4)(V (R) V (R/2)). R (27) V (R/2) V (R/4) V (R/4) (V (R) V (R/2)), C for some universal constant C. The inequality holds for any R > 4, in particular, for any k 0, we have V (2 k R) V (2 k 2 R) V (2k 2 R) (V (2 k R) V (2 k R)) C (28) V (R/4) (V (2 k R) V (2 k R)). C Choose C 0 = 2C. Take R = 4R 0 > 4, we then have for any k 0, k ( V (2 k R) V (R/2) = V (2 i R) V (2 i R) ) (29) (28) i=0 k i=0 ( ) i+ C (V (R/2) V (R/4)) V (R/4) V (R/2) V (R/4), where in the last inequality, we have used V (R/4) V (R 0 ) C 0 = 2C. Therefore (30) 2V (2R 0 ) = 2V (R/2) V (2 k R) = for any k. The lemma is proved by taking k. 0 B 2 k R ϕ 2

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