On the relation between scaling properties of functionals and existence of constrained minimizers

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1 On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

2 The minimization of energy functionals under some suitable constraint is a classical problem in mathematical physics. Some example: Thomas-Fermi energy for molecular systems where E TF (u) := 1 2 V(x) = R 3 R 3 u(x)u(y) x y dxdy + γ k z j x R j 1, U = j=1 The central problem is to compute u 5/3 dx uv(x)dx + U R 3 R 3 1 i<j k E TF = inf B λ E TF (ψ) where B λ := {u 0, u L 1 L 5/3 R 3 u(x)dx = λ}. z i z j R i R j 1 J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

3 Thomas-Fermi-Dirac energy for molecular systems E TFD (u) := E TF (u) c u(x) 4/3 dx R 3 Thomas-Fermi-Von Weizsacker energy E TFW (u) := E TF (u) + d R 3 u 2 dx Thomas-Fermi-Dirac-Von Weizsacker energy E TFDW (u) := E TF (ψ) c R 3 u(x) 4/3 dx + d R 3 u 2 dx Some references Lieb-Simon (1977) for Thomas-Fermi theory, Benguria-Brezis-Lieb(1981) for Thomas-Fermi-Von Weizsacker theory, Benguria (1979) and Lieb (1981) for Thomas-Fermi-Dirac theory, Lions (1987) and Catto-Lions (1992) for Thomas-Fermi-Dirac-Von Weizsacker theory. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

4 The free electron gas: the Schrödinger-Poisson-Slater equation (SPS). iψ t + ψ ( x 1 ψ 2 )ψ + ψ p 2 ψ = 0 in R 3, p = 8 3. Standing waves solutions for (4) ψ(x, t) = u(x)e iωt can be found minimizing I(u) = 1 u 2 dx + 1 R3 R3 u(x) 2 u(y) 2 dxdy 1 u p dx 2 R 3 4 x y p R 3 under the constraint B ρ = {u H 1 (R 3 ; C) : u 2 = ρ}. What is known for p = 8. Sanchez-Soler J. Stat. Phys. (2004) 3 Called I ρ 2 = inf B ρ I(u), there exists ρ 1 > 0 such that for all 0 < ρ < ρ 1 the infimum is achieved. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

5 Open problem The Cauchy probelm for SPS equation in case 2 < p < 10 3 is well studied: global existence for the evolution equation for H 1 data. What about the minimization problem? Do exist standing waves with arbitrary L 2 norm in case 2 < p < 10 3? Abstract minimization Problem We look for minimizers of the following problem I ρ 2 = inf B ρ I(u) where B ρ := {u H 1 (R N ) such that u 2 = ρ} and I(u) := 1 2 u 2 D 1,2 + T(u) (1) J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

6 Some comment We don t look for radial solution. The problem is translation invariant. Standard problems for translation invariant functionals It is known that, in this kind of problems, that main difficulty concerns with the lack of compactness of the (bounded) minimizing sequences {u n } B ρ ; indeed two possible bad scenarios are possible: u n 0 (vanishing); u n ū 0 and 0 < ū 2 < ρ (dichotomy). The main contribution to constained minimization is due to the concentration-compactness thorems by P.L. Lions. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

7 Theorem A [B., Bonanno, Proc. Roy. Soc. Edimb. sec A (2010)] Let T be a C 1 functional on H 1 (R N ). Let ρ > 0 and {u n } be a minimizing sequence for I ρ 2 weakly convergent, up to translations, to a nonzero function ū. Assume the strong subadditivity inequality and that I ρ 2 < I µ 2 + I ρ 2 µ 2 for any 0 < µ < ρ. T(u n ū) + T(ū) = T(u n ) + o(1); (2a) T(α n (u n ū)) T(u n ū) = o(1) where α n = ρ2 ū 2 2 u n ū 2 ; < T (u n ), u n >= O(1); (2b) (2c) < T (u n ) T (u m ), u n u m >= o(1) as n, m +. (2d) Then u n ū H 1 0. In particular it follows that ū B ρ and I(ū) = I ρ 2. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

8 Theorem [B., Siciliano (2010) Z. Angew. Math. Phys. (2010)] Let p (3, 10 3 ). Then there exist ρ 1 > 0 (depending on p) such that all the minimizing sequences are precompact in H 1 (R 3 ; C) up to translations provided that ρ 1 < ρ < + if 3 < p < In particular there exists a couple (ω ρ, u ρ ) R H 1 (R 3 ; R) solution of SPS. Some comments For 3 < p < 10 3 we are able to show the weakly convergence of the minimizing sequences to a nonzero function only for large ρ. The strong subadditivity inequality holds for large ρ. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

9 The weak subadditivity inequality Lions proved that the invariance by translations of the problem implies in many cases an inequality that the infima I ρ 2 have to satisfy and read as follows: weak subadditivity inequality I ρ 2 I µ 2 + I ρ 2 µ 2 for all 0 < µ < ρ. (3) The strong subadditivity inequality The crucial strong subadditivity inequality I ρ 2 < I µ 2 + I ρ 2 µ 2 for any 0 < µ < ρ. holds when the function s I s 2 in the interval [0,ρ] achieves its unique s 2 minimum in s = ρ. Indeed for µ (0,ρ) we get µ2 I ρ 2 ρ 2 < I µ 2 and ρ2 µ 2 I ρ 2 ρ 2 < I ρ 2 µ 2. Therefore I ρ 2 = µ2 ρ 2 I ρ 2 + ρ2 µ 2 ρ 2 I ρ 2 < I µ 2 + I ρ 2 µ 2 µ (0,ρ). J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

10 Monotonicity In order to achieve the strong subadditivity for any ρ > 0 it is sufficent that the function s I s 2 is monotone decreasing. s 2 Lemma Assume that for every ρ > 0, < I ρ 2 s I s 2 is continuous (4) lim s 0 I s 2 = L. (5) s2 I s 2 < L for small s 0. (6) s2 Then there exists a sequence ρ n 0 such that for all n I ρ 2 n < I µ 2 + I ρ 2 n µ 2 for all 0 < µ < ρ n. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

11 Bad news: monotonicity is not for free The function s I s 2 can have a fast oscillating behavior, even in a s 2 neighborhood of the origin, even if the function s I s 2 is continuous and fulfills the weak subadditivity inequality (3). Example: the Cantor function, Dobos Proc. Amer. Math. Soc. (1996). Some definitions Let u H 1 (R N ), u 0. A continuous path g u : θ R + g u (θ) H 1 (R N ) such that g u (1) = u is said to be a scaling path of u if g u (θ) 2 2 = H gu (θ) u 2 2 with H gu differentiable and H g u (1) 0 (7) where the prime denotes the derivative. We denote with G u the set of the scaling paths of u. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

12 Scaling paths The set G u is nonempty and indeed it contains a lot of elements: for example, g u (θ) = θu(x) G u, since H gu (θ) = θ 2. Also g u (θ) = u(x/θ) is an element of G u since H gu (θ) = θ N. As we will see in the application it is relevant to consider the family of scaling paths of u parametrized with β R given by G β u = {g u(θ) = θ 1 N 2 β u(x/θ β )} G u. (8) Notice that all the paths of this family have as associated function H(θ) = θ 2. This allows us to pass from one constraint to another one by having one degree of freedom (the parameter β). J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

13 Theorem C [B., Siciliano (2010)] Assume that for every ρ > 0, all the minimizing sequences u n for I ρ 2 have a weak limit, up to translations, different from zero. Assume hypoteses (2), (4), (5) and (6) then there exists a sequence ρ n 0 such that for all n there exists minimizers in B ρn. Moreover if where u minimizer g u G u such that h gu (θ) := I(g u (θ)) H gu (θ)i(u) d dθ h g u (θ) θ=1 0, (9) then I s 2 is monotone decreasing. Moreover, if {u s 2 n } is a minimizing sequence weakly convergent to a certain ū ( 0) then u n ū H 1 0 and I(ū) = I ρ 2. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

14 Idea of the proof I Let us fix ρ > 0 and call c := min s 2 [0,ρ] < L, by (6). s 2 { ρ 0 := min s [0,ρ] s.t I } s 2 s 2 = c. The function s I s 2 in the interval [0,ρ s 2 0 ] achieves the minimum only in s = ρ 0. Hence there exists a minimizer ū in B ρ0. We have to prove that ρ 0 = ρ. Now we argue by contradiction by assuming that ρ 0 < ρ. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

15 Idea of the proof Fixed gū Gū with its associated H and the definition of ρ 0 : I ρ 2 0 ρ 2 0 Therefore we have I H(θ)ρ 2 0 H(θ)ρ 2 0 for all θ (1 ε, 1 + ε). I(gū(θ)) H(θ)ρ 2 0 I H(θ)ρ 2 0 H(θ)ρ 2 0 I ρ 2 0 ρ 2 0 = I(ū) ρ 2 0 for every θ (1 ε, 1 + ε). J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

16 Idea of the Proof This means that the map h gū(θ) = I(gū(θ)) H(θ)I(ū), defined in a neighborhood of θ = 1, is non negative and has a global minimum in θ = 1 with h gū(1) = 0. Then, denoting with a prime the derivative of h gū with respect to θ, we get h gū(1) = 0. Since gū is arbitrary this relation has to be true for every map gū, so we have found a minimizer ū such that for every gū Gū, h gū(1) = 0: this clearly contradicts (9) and so ρ 0 = ρ. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

17 Existence of standing waves for Schrödinger-Poisson-Slater systems for 2 < p < 3 I(u) = 1 u 2 dx + 1 R3 R3 u(x) 2 u(y) 2 dxdy 1 u p dx 2 R 3 4 x y p R 3 under the constraint B ρ = {u H 1 (R 3 ; C) : u 2 = ρ} with 2 < p < 3. We define, for short, the following quantities: R3 R3 A(u) = u 2 u(x) 2 u(y) 2 dx, B(u) = dxdy, C(u) = u p dx. R 3 x y R 3 so that I(u) = 1 2 A(u) B(u) + 1 p C(u). Behavior near zero I lim s 2 s 0 s 2 = 0, I s 2 < 0 s, weak limit 0 s2 J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

18 Existence of standing waves for Schrödinger-Poisson-Slater systems for 2 < p < 3 I(u) = 1 u 2 dx + 1 R3 R3 u(x) 2 u(y) 2 dxdy 1 u p dx 2 R 3 4 x y p R 3 under the constraint B ρ = {u H 1 (R 3 ; C) : u 2 = ρ} with 2 < p < 3. We define, for short, the following quantities: R3 R3 A(u) = u 2 u(x) 2 u(y) 2 dx, B(u) = dxdy, C(u) = u p dx. R 3 x y R 3 so that I(u) = 1 2 A(u) B(u) + 1 p C(u). Behavior near zero I lim s 2 s 0 s 2 = 0, I s 2 < 0 s, weak limit 0 s2 J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

19 The scaling laws Let u H 1 (R 3, C) and choose the family of scaling paths given by g u (θ) = θ β u(x/θ β ) such that H(θ) = θ 2. We easily find the following scaling laws: A(g u (θ)) = θ 2 2β A(u), B(g u (θ)) = θ 4 β B(u), C(g u (θ)) = θ (1 3 2 β)p+3β C(u). We get h gu (θ) = 1 2 (θ2 2β θ 2 )A(u)+ 1 4 (θ4 β θ 2 )B(u)+ 1 p (θ(1 3 2 β)p+3β θ 2 )C(u). J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

20 Computation of h g u (1) We compute h g u (1). Assume that h g u (1) = 0 for all g u G β u implies B(u) = 2A(u), C(u) = I(u) = A(u) 2 + B(u) 4 p 2 p A(u), + C(u) p = 3 p 2 p A(u). (10) We show that relations (10) are impossible for p (2, 3) for small ρ by means of Hardy-Littlehood-Sobolev inequality and interpolation inequality. J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

21 Example in case 2 < p < 12 5 B(u) c u 4 12/5 c u 4α p u 4(1 α) 6, α = 3p 2(6 p). We get, thanks to (10) and the Sobolev inequality u 2 6 SA(u), B(u) cb(u) 4α p B(u) 4(1 α) 2. (11) We notice that 4α p + 4(1 α) 2 > 1 since p < 3. This is in contradiction with B(u) 0 when ρ 0. Example in case p = 8 3 (the case of Sanchez-Soler (2004)). B(u) c u 4/3 2 u = cρ 4/3 C(u) J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

22 Theorem [B., Siciliano (2010)] Let p (2, 3). Then there exists ρ 1 (depending on p) such that all the minimizing sequences are precompact in H 1 (R 3 ; C) up to translations provided that 0 < ρ < ρ 1. Hence that for all 0 < ρ < ρ 1 the infimum is achieved. Orbital stability Following Cazenave-Lions (1983), the set is orbitally stable. S ρ = {e iθ u(x) : θ [0, 2π), u 2 = ρ, I(u) = I ρ 2}, J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

23 Next steps Monotonicity of s I s 2 s 2 for large s for 2 < p < 3? Existence of solitary wave solution of the following relativistic nonlinear Schrödinger equation iψ t = 1 ψ ψ + ( x 1 ψ 2 )ψ ψ p 2 ψ The minimization problem in H 1/2 is interesting in case 2 < p 8 3 (the Slater exponent is 8 3 ). Some problems: local and global existence of the evolution equation, the existence of minimizers ( in case of attractive Coulombic interaction Lieb-Yau (1987), Frohlich et al. without radial assumption (2007)). J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23

24 References J. Bellazzini, N. Visciglia, On the orbital stability for a class of nonautonomous NLS, Indiana Univ. Math. J. 59 (2010), n. 3., J. Bellazzini, C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials, Proc. Roy. Soc. Edimburgh sec A 140 (2010), J. Bellazzini, G. Siciliano, Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys, (2010) to appear (arxiv: ) J. Bellazzini, G. Siciliano, On the relation between scaling properties of functionals and existence of constrained minimizers, Preprint J. Bellazzini (DMA) Scaling properties and existence of minimizers January 11, / 23