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Magnetic rings
Jean Dolbeault, Maria J. Esteban, Ari Laptev, Michael Loss
To cite this version:
Jean Dolbeault, Maria J. Esteban, Ari Laptev, Michael Loss. Magnetic rings. Journal of Mathematical
Physics, American Institute of Physics (AIP), 2018, 59 (5), pp.051504. �10.1063/1.5022121�.
�hal-01680917v2�
Jean Dolbeault,1 Maria J. Esteban,1Ari Laptev,2and Michael Loss3
1)
CEREMADE (CNRS UMR n◦ 7534), PSL research university, Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, Francea)
2)
Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, UKb)
3)School of Mathematics, Skiles Building, Georgia Institute of Technology, Atlanta GA 30332-0160,
USAc)
(Dated: 18 May 2018)
We study functional and spectral properties of perturbations of the operator −(∂s+ i a)2 in L2(S1). This
operator appears when considering the restriction to the unit circle of a two dimensional Schr¨odinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on R2
and, on S1, a sharp
interpolation inequality and a sharp Keller-Lieb-Thirring inequality. PACS numbers: 02.30.-f; 02.30.Hq; 02.30.Xx; 02.60.Lj
I. INTRODUCTION
On the two-dimensional Euclidean space R2, let us
in-troduce the polar coordinates (r, ϑ) ∈ [0, +∞) × S1 of
x ∈ R2and consider a magnetic potential a in a
transver-sal gauge, or Poincar´e gauge1, so that (a, e
r) = 0 and
(a, eϑ) = aϑ(r, ϑ), where (er, eϑ) is the oriented
orthogo-nal basis associated with the polar coordinates such that, for any x ∈ R2\ {0}, e
r= x/r, r = |x|. With this
nota-tion, the energyR
R2|(i ∇+a) Ψ|
2dx corresponding to the
magnetic Schr¨odinger operator −∆acan be rewritten as
Z +∞ 0 Z π −π |∂rΨ|2+ 1 r2| ∂ϑΨ + i r aϑΨ| 2 r dϑ dr . One of the main motivations is the study of Bohm-Aharonov magnetic fields2,3with aϑ(r, ϑ) = a/r for some
constant a ∈ R. We recall that Stokes’ formula applied to the magnetic field b = curl a shows that the magnetic flux is given by Z |x|<r b dx = 1 2π Z π −π aϑ(r, ϑ) r dϑ = a .
The main result concerning Bohm-Aharonov magnetic fields is, for an arbitrary non-negative function ϕ in Lq
(S1), q ∈ (1, +∞), the Hardy-type inequality
Z R2 |(i ∇ + a) Ψ|2dx ≥ τ Z R2 ϕ(x/|x|) |x|2 |Ψ| 2dx (1)
which holds for some constant τ depending on kϕkLq(S1).
A precise statement will be given in CorollaryII.3. The proof relies on a method4 developed recently and
a)Electronic mail: dolbeaul@ceremade.dauphine.fr; Electronic mail:esteban@ceremade.dauphine.fr
b)Also at Department of Mathematics, Siberian Federal University, Russia ; Electronic mail: a.laptev@imperial.ac.uk
c)Electronic mail:loss@math.gatech.edu
uses a Keller-Lieb-Thirring inequality for the first eigen-value of a magnetic Schr¨odinger operator on a magnetic ring (see CorollaryII.2). This spectral estimate is equiv-alent to sharp interpolation inequalities for a magnetic Laplacian on the circle and has been inspired by a series of previous papers5–7 on interpolation inequalities and
their spectral counterparts. Let us mention that some semiclassical properties of the spectrum of magnetic rings were recently studied including an electric potential that admits a double symmetric well8 (also see earlier
refer-ences therein). Our results are not limited to the semi-classical regime.
II. MAIN RESULTS
On (−π, π] ≈ S1, let us consider the uniform proba-bility measure dσ = ds/(2π) and denote by kψkLp(S1)
the corresponding Lp norm, for any p ≥ 1. Assume that
a : R → R is a 2π-periodic function such that its restric-tion to (−π, π] ≈ S1 is in L1
(S1) and define the subspace
Xa:=ψ ∈ Cper(R) : ψ0+ i a ψ ∈ L2(S1)
of the space Cper(R) of the continuous 2π-periodic
func-tions on R. The change of function ψ(s) 7→ eiR−πs (a(s)−¯a)dσψ(s) ,
where ¯a :=R−ππ a(s) dσ is the magnetic flux, reduces the problem to the case of a constant: in the sequel of this paper we shall always assume that
a is a constant function. Replacing ψ by s 7→ eiks
ψ(s) for any k ∈ Z shows that µa,p(α) = µk+a,p(α) so that we can restrict the problem
to a ∈ [0, 1]. By considering χ(s) = e−isψ(s), we find |ψ0+ i a ψ|2= |χ0+ i (1 − a) χ|2=
ψ0− i a ψ
2
2 and thus µa,p(α) = µ1−a,p(α): it is thus enough to
con-sider the case a ∈ [0, 1/2].
Using a Fourier series ψ(s) =P
k∈Zψke iks, we obtain that kψ0+ i a ψk2L2(S1)= X k∈Z (a + k)2|ψk|2≥ a2kψk2L2(S1), so that ψ 7→ kψ0+ i a ψk2 L2(S1)+α kψk2L2(S1)is coercive for
any α > − a2. Moreover, the optimal constant µ a,p(α)
in the interpolation inequality
kψ0+ i a ψk2L2(S1)+ α kψk2L2(S1)≥ µa,p(α) kψk2Lp(S1) (2)
written for any ψ ∈ Xa is an increasing concave function
of α > − a2characterized by µa,p(α) := inf ψ∈Xa\{0} Rπ −π |ψ 0+ i a ψ|2+ α |ψ|2 dσ kψk2 Lp(S1) (3) and7lim
α→− a2µa,p(α) = 0. The inequality (2) is known
if either p = +∞9,10or p = − 211and the expression of an
optimal function was given as a series9for any α > − a2
when p = +∞. Our first result is the extension of this interpolation result to the case p ∈ (2, +∞).
Theorem II.1 For any p > 2, a ∈ R, and α > − a2, the infimum in (3) is achieved and
(i) if a ∈ [0, 1/2] and a2(p + 2) + α (p − 2) ≤ 1, then µa,p(α) = a2+ α and equality in (2) is achieved
only by the constant functions,
(ii) if a ∈ [0, 1/2] and a2(p + 2) + α (p − 2) > 1, then µa,p(α) < a2+ α and equality in (2) is not achieved
by the constant functions. Moreover, for any α > − a2, a 7→ µ
a,p(α) is monotone
increasing on (0, 1/2).
More can be said on µa,p(α): see Theorem III.7. The
region a2(p + 2) + α (p − 2) < 1 is exactly the set where
the constant functions are linearly stable critical points. See Figs.1and2.
With the results of Theorem II.1 in hand, we study some spectral properties of the magnetic Schr¨odinger op-erator Ha− ϕ on the unit circle S1≈ (−π, π] 3 s where ϕ
is a potential and Ha is the magnetic Laplacian given by
Haψ(s) = −
d ds+ i a
2 ψ(s) .
The presence of a non-trivial magnetic field a in Ha
“lifts” the spectrum up and the final result substantially depends on its value. Note that Lieb-Thirring inequali-ties with magnetic field10, in particular, imply an
inequal-ity for the first eigenvalue. However, it is not known if the constant is sharp. A somewhat similar result where the lifting of the spectrum is provided by a constant mag-netic field was proved with different methods7.
The first spectral consequence of Theorem II.1 is a Keller-Lieb-Thirring inequality for the first eigenvalue λ1(Ha − ϕ) of the Schr¨odinger operator Ha− ϕ. The
function α 7→ µa,p(α) is monotone increasing, concave,
and therefore has an inverse, denoted by αa,p : R+ →
(−a2, +∞), which is monotone increasing, and convex. Corollary II.2 Let p > 2, a ∈ [0, 1/2], q = p/(p − 2) and assume that ϕ is a non-negative function in Lq(S1). Then
λ1(Ha− ϕ) ≥ − αa,p kϕkLq(S1) . (4)
If 4 a2+ µ (p − 2) ≤ 1, then α
a,p(µ) = µ − a2; if 4 a2+
µ (p − 2) > 1, then αa,p(µ) > µ − a2.
These estimates are optimal in the sense that there ex-ists a non-negative function ϕ such that λ1(Ha− ϕ) =
− αa,p kϕkLq(S1). If 4 a2+ µ (p − 2) ≤ 1, then the
equal-ity in (4) is achieved by constant potentials.
The second application of Theorem II.1 is related to a Hardy inequality in R2. Let us consider the Bohm-Aharonov vector potential
a(x) = a x2 |x|2, −x1 |x|2 , x = (x1, x2) ∈ R2, a ∈ R .
and recall the inequality3 Z R2 |(i ∇ + a) Ψ|2dx ≥ min k∈Z(a − k) 2Z R2 |Ψ|2 |x|2 dx . (5)
Using interpolation inequalities5, the following version4
of Hardy’s inequality in the case d ≥ 3 was proved: Z Rd |∇Ψ|2dx ≥ τ Z Rd ϕ(x/|x|) |x|2 |Ψ| 2dx ,
where the constant τ depends on the value of kϕkLq(Sd−1).
Using similar arguments we are now able to prove the following result.
Corollary II.3 Let p > 2, a ∈ [0, 1/2], q = p/(p − 2) and assume that ϕ is a non-negative function in Lq(S1). Then Inequality (1) holds with τ > 0 being the unique solution of the equation
αa,p(τ kϕkLq(S1)) = 0 .
Moreover, τ = a2/kϕk
Lq(S1)if 4 a2+kϕkLq(S1)(p−2) ≤ 1.
Notice that for any a ∈ (0, 1/2), by taking ϕ constant, small enough in order that 4 a2+ kϕk
Lq(S1)(p − 2) ≤ 1,
we recover the inequality Z R2 |(i ∇ + a) Ψ|2dx ≥ a2 Z R2 |Ψ|2 |x|2 dx ,
which is a equivalent to (5). The case a = 1/2 is obtained by a limiting procedure and for arbitrary values of a ∈ R, we refer to the observations of SectionIII.
III. PROOF OF THEOREMII.1AND FURTHER RESULTS
Lemma III.1 For all a ∈ R, p ∈ (2, ∞) and α ≥ −a2,
equality in (2) is achieved by at least one function in Xa.
Indeed, by the diamagnetic inequality
|ψ|0
≤ |ψ0+ i a ψ| a.e. ,
which holds for any ψ ∈ Xa, we infer that any
mini-mizing sequence {ψn} for (3) can be taken bounded in
H1
(S1). By the compact Sobolev embeddings, this
se-quence is relatively compact in Lp
(S1 ) and in C(S1). The maps ψ 7→Rπ −π|ψ| 2dσ and ψ 7→Rπ −π|ψ 0+ i a ψ|2dσ are
lower semicontinuous by Fatou’s lemma, which proves the
claim.
The minimization problem (3) has several reformula-tions, that have already been used in the case α = 012.
1) Any solution ψ ∈ Xa of the minimization problem (3)
satisfies the Euler-Lagrange equation (Ha+ α) ψ = |ψ|p−2ψ
up to a multiplication by a constant. We observe that v(s) = ψ(s) eias satisfies the condition
v(s + 2π) = e2iπav(s) ∀ s ∈ R , (6) and we can reformulate (3) as
µa,p(α) = min v∈Ya\{0}
Qp,α[v]
where Ya:= {v ∈ C(R) : v0∈ L2(S1) , (6) holds} and
Qp,α[v] := kv0k2 L2(S1)+ α kvk2L2(S1) kvk2 Lp(S1) .
2) With v = u eiφ written in polar form, the boundary
condition becomes
u(π) = u(−π) , φ(π) = 2π (a + k) + φ(−π) (7) for some k ∈ Z, and kv0k2
L2(S1)= ku0k 2 L2(S1)+ku φ0k 2 L2(S1). We can reformulate (3) as µa,p(α) = min (u,φ)∈Za\{0} ku0k2 L2(S1)+ ku φ0k2L2(S1)+ α kuk2L2(S1) kuk2 Lp(S1) where Za := {(u, φ) ∈ C(R)2 : u0, u φ0∈ L2(S1) , (7) holds}.
3) The third reformulation of (3) relies on the Euler-Lagrange equations
− u00+ |φ0|2u + α u = |u|p−2u and (φ0u2)0 = 0 .
Integrating the second equation, and assuming that u
never vanishes, we find a constant L such that φ0= L/u2. Taking (7) into account, we deduce from
L Z π −π ds u2 = Z π −π φ0ds = 2π (a + k) that ku φ0k2 L2(S1)= L2 Z π −π dσ u2 = (a + k)2 ku−1k2 L2(S1) . Hence φ(s) − φ(0) = a + k ku−1k2 L2(S1) Z s −π ds u2. Let us define Qa,p,α[u] := ku0k2 L2(S1)+ a2ku−1k −2 L2(S1)+ α kuk2L2(S1) kuk2 Lp(S1) .
In what follows, we denote by H1
(S1) the subspace of
the continuous functions u on (−π, π] such that u(π) = u(−π) and u0∈ L2
(S1). Notice that if u ∈ H1(S1) is such that u(s0) = 0 for some s0∈ (−π, π], then
|u(s)|2= Z s s0 u0ds 2 ≤√2π ku0kL2(S1)p|s − s0|
and u−2 is not integrable. In this case we adopt the convention that Qa,p,α[u] = Qp,α[u].
Lemma III.2 For any a ∈ (0, 1/2), p > 2, α > − a2,
µa,p(α) = min
u∈H1(S1)\{0}Qa,p,α[u]
is achieved by a function u > 0.
To prove this result, it is enough to check that the in-fimum (3) is achieved by a function ψ ∈ Xa such that
ψ(s) 6= 0 for any s ∈ (−π, π]. Without loss of generality, we can assume that ψ is an optimal function for (2) with kψkLp(S1) = 1. Let us decompose v(s) = ψ(s) eias as a
real and an imaginary part, v = v1+ i v2, which both
solve the same Euler-Lagrange equation − vj00+ α vj= (v12+ v
2 2)
p
2−1vj, j = 1 , 2 .
The Wronskian w = (v1v20 − v01v2) is constant.
Neither v1 nor v2 vanishes identically on S1 because
of (6). If both v1 and v2 vanish at the same point, then
w vanishes identically, which means that v1 and v2 are
proportional. Again, this cannot be true because of the twisted boundary condition (6). If a = 0, Qa=0,p,α[u] = Qp,α[u] for any u ∈ H1(S1)\{0}.
Lemma III.3 For any p > 2, if 0 < α ≤ 1/(p − 2), then µ0,p(α) = α is achieved only by constant functions.
4 2) = −1/4 = µ0,p(−1/4), with equality achieved only by
constant functions.
Both results (case p > 2,5,13and case p = − 2,11) were al-ready known. For p > 2, similar results have been found by various methods based on entropy techniques14–17and
the carr´e du champ method of D. Bakry and M. Emery18.
There are many earlier references19–25 on Kolmogorov’s
inequalities26 corresponding to various other boundary
conditions and intervals.
As a consequence of the cases p > 2 and p = − 2 we have the inequalities
ku0k2
L2(S1)+β kuk2L2(S1)≥ β kuk2Lp(S1) ∀ u ∈ H1(S1) (8)
for any p > 2 and β ∈ (0, 1/(p − 2)], and ku0k2 L2(S1)+ 1 4ku −1k2 L2(S1)≥ 1 4kuk 2 L2(S1) ∀ u ∈ H1(S1) . (9) Inequality (9) actually enters in the family of inequali-ties (8), with the parameter β = −1/4 = 1/(p − 2) corre-sponding to the critical exponent p = 2 d/(d − 2) = − 2 since here d = 1. This exponent is critical from the point of view of scalings because, at least for a function u with compact support in (−π, π), kukLp(S1) scales like
ku0k
L2(S1). This is why a unified proof of both cases can
be done with the Bakry-Emery method: see AppendixA. We are now ready to study the key issues of TheoremII.1. Lemma III.4 Let p > 2, a ∈ [0, 1/2], and α > − a2.
(i) if a2(p+2)+α (p−2) ≤ 1, then µa,p(α) = a2+α and
equality in (2) is achieved only by the constants, (ii) if a2(p + 2) + α (p − 2) > 1, then µ
a,p(α) < a2+ α
and equality in (2) is not achieved by the constants. In case (i), we can write
ku0k2 L2(S1)+ a2ku−1k−2L2(S1)+ α kuk 2 L2(S1) = (1 − 4 a2) ku0k2L2(S1)+ α kuk 2 L2(S1) + 4 a2 ku0k2 L2(S1)+ 1 4ku −1k2 L2(S1)
and conclude using (9) and then (8) with β = a
2+ α
1 − 4 a2 ≤
1 p − 2.
In case (ii), let us consider the test function uε :=
1 + ε w1, where w1 is the eigenfunction corresponding
to the first non zero eigenvalue of −d2/ds2 on H1
(S1),
with Neumann boundary conditions, namely, λ1= 1 and
w1(s) = 1 + cos s. A Taylor expansion shows that
Qa,p,α[uε] = a2+α+ 1−a2(p+2)− α (p−2) ε2+o(ε2) ,
which proves the result.
The proof of Lemma III.4, (i) relies on (8) and (9). It is remarkable that it does not use rigidity results based on the carr´e du champ method, at least directly. Notice that results similar to LemmaIII.4were known for p = +∞9,10 using a Fourier representation of the operator
and for an arbitrary p > 2 if α = 012.
It follows from the definition of Qa,p,α[u] that a 7→
µa,p(α) is nondecreasing on [0, 1/2). The strict
mono-tonicity follows from the existence of an optimal func-tion, which is known by Lemma III.1. This concludes the proof of TheoremII.1. The remainder of this section is devoted to complementary results, which specify the range of µa,p(α) when a varies in [0, 1/2).
Let us consider νp(α) := inf v∈H1 0(S1)\{0} Qp,α[v] . Here H1
0(S1) denotes the subspace of the functions v ∈
H1
(S1) such that v(±π) = 0. Since (6) is satisfied by any
function in H1
0(S1), we have the following estimate.
Lemma III.5 If p > 2, α > − a2
and a ∈ R, then µa,p(α) ≤ νp(α) .
Moreover, this inequality is strict if a ∈ [0, 1/2).
If {un}n∈Nis a minimizing sequence such that, for any
n ∈ N, kunkLp(S1) = 1, then it is clearly bounded in
H1
(S1), and so, by the compact Sobolev embeddings, it
is relatively compact in L2
(S1), Lp
(S1
) and C(S1). Up
to subsequences, {un}n∈N converges to some function u
weekly in H1 and strongly in L2(S1), Lp(S1) and C(S1). After noticing that Qp,α[ |u| ] = Qp,α[u], we obtain the
following result.
Lemma III.6 If p > 2, α > − a2, then νp(α) admits a
non-negative minimizer.
The strict monotonicity of a 7→ µa,p(α) is a consequence
of LemmaIII.6and, as a consequence, we know that µa,p(α) < µ1/2,p(α) ≤ νp(α)
for any a ∈ [0, 1/2). It turns out that the last inequality is an equality.
Theorem III.7 For any p > 2 and α > − a2, we have µ1/2,p(α) = νp(α) .
This result was already known for the limit cases p = 2,11 and p = +∞,9,10. To prove it, we set v(s) = eis/2ψ(s)
and note that v(s + 2π) = − v(s) for all s, which follows from the periodicity condition (6) with a = 1/2. More-over, the derivative v0satisfies v0(s + 2π) = − v0(s). Note
that these boundary conditions also hold for the real part and the imaginary part of v separately. We call them v1
these conditions. Both v1 and v2 must vanish at some
point but a priori these points need not be the same. We set ηj = |vj|, j = 1, 2, and note that
Qp,α[v] = Rπ −πη021 + η202 dσ + α R π −πη 2 1+ η22 dσ kηk2 p .
The functions ηj are now periodic. They are not
neces-sarily smooth but are at least continuous. Now we re-place both η1and η2by their symmetric decreasing
rear-rangements around the point 0. The numerator decreases for the usual reasons and the denominator increases (see Lemma B.1, in Appendix B). Thus, the symmetrically decreasing rearranged functions η1∗ and η2∗ have a maxi-mum at 0 and vanish at ±π, so that η∗1+ i η∗2 ∈ H1
0(S1).
If v is a minimizer of Qp,α under Condition (6) with
a = 1/2, then
νp(α) ≤ Qp,α[η∗1+ i η2∗] ≤ Qp,α[v] = µ1/2,p(α) .
With the convention that Qa,p,α[u] = Qp,α[u] if u ∈
H1
0(S1), we can claim that the infimum of Qa,p,α is
at-tained by some u ∈ H1
(S1) \ {0} for any a ∈ [0, 1/2],
including in the case a = 1/2 for which the minimizer can be taken in H10(S1) \ {0}.
IV. PROOF OF COROLLARIESII.2ANDII.3
Let us start with the proof of Corollary II.2. Con-sider the quadratic form associated with Ha− ϕ. Using
H¨older’s inequality, we obtain
kψ0+ i a ψk2L2(S1)− Z π −π ϕ |ψ|2dσ ≥ kψ0+ i a ψk2L2(S1)− µ kψk2Lp(S1) where µ = kϕkLq(S1) and 1 q + 2 p = 1. Let us choose α
such that µa,p(α) = µ. It follows from (2) that
kψ0+ i a ψk2L2(S1)− µ kψkL2p(S1)≥ − α kψk2L2(S1)
and from TheoremII.1that µa,p(α) = a2+ α if a2(p +
2) + α (p − 2) ≤ 1. This implies that λ1(Ha− ϕ) ≥ a2− kϕkLq(S1)
if 4 a2+ kϕkLq(S1)(p − 2) ≤ 1. In that case the equality
is achieved by ϕ ≡ const. The proof is complete.
Now let us prove CorollaryII.3. Let x = (r, ϑ) ∈ R2
be polar coordinates in R2. Then we find Z R2 |(i ∇ + a) Ψ|2dx = Z ∞ 0 Z S1 r |∂rΨ|2+ 1 r|∂ϑΨ + i a Ψ| 2 dϑ dr . Let τ > 0. Then Z ∞ 0 Z S1 1 r |∂ϑΨ + i a Ψ| 2− τ ϕ |Ψ|2 dϑ dr ≥ λ1(Ha− τ ϕ) Z ∞ 0 Z S1 1 r|Ψ| 2 dϑ dr ≥ − αa,p(τ kϕkLq(S1)) Z ∞ 0 Z S1 1 r|Ψ| 2dϑ .
Note that if τ = 0, then
αa,p(τ kϕkLq(S1)) = αa,p(0) = − a2,
and for a sufficiently large τ the value of αa,p(τ kϕkLq(S1))
is positive. Therefore we can find τ > 0 such that αa,p(τ kϕkLq(S1)) = 0. This value is unique since αa,p(µ)
is strictly monotone with respect to µ. The conclusion
easily follows.
Appendix A: A proof of LemmaIII.3by the carr´e du champ method
Let Fβ[u] := ku0k2L2(S1)+ β kukL22(S1)− kuk2Lp(S1). If
p > 2, it is enough to prove µ0,p(β) = β for β = α?,
α?:= 1/(p − 2), because
Fβ[u] = 1 − β (p − 2) ku0k2L2(S1)+ β (p − 2) Fα?[u]
if 0 < β ≤ α?. Let us consider a positive solution of the
parabolic equation ∂u ∂t = u 00+ (p − 1)|u0|2 u and compute − d dtFα?[u(t, ·)] = Z π −π |u00|2− |u0|2 dσ +p − 1 3 Z π −π |u0|4 u2 dσ
using several integrations by parts. The first term in the r.h.s. is non-negative by the Poincar´e inequality, as well as the second one. Notice that ρ = |u|p is a solution
of the heat equation, so that positivity is preserved by the flow and Fα?[u(t = 0, ·)] ≥ limt→+∞Fα?[u(t, ·)] = 0,
which is exactly (8) written with u = u(t = 0, ·). The strict positivity condition is easily removed by an approx-imation procedure. Exactly the same computations give
6 the result in the case p = − 2 and establish (9).
For p > 2, the method is well known17,18. The result
for p = − 2 was established earlier11 but, as far as we
know, this proof is new.
Appendix B: A symmetrization result
Here f∗ denotes the symmetric decreasing rearrange-ment of f .
Lemma B.1 Let p ≥ 2. For any non-negative functions f , g ∈ Lp(S1) we have that Z π −π f2+ g2p/2 dσ ≤ Z π −π f∗2+ g∗2 p/2 dσ . The case p = 2 is obvious, in fact there is equality. Hence we assume that p > 2. Write
Z π −π f2+ g2p/2dσ 2/p = sup kvkLq (S1)=1 Z π −π f2+ g2 v dσ
where 1/q + 2/p = 1. Clearly, we may choose v to be positive. By standard rearrangement inequalities,
Z π −π f2+ g2 v dσ ≤ Z π −π f∗2+ g∗2v∗dσ and kv∗k
Lq(S1)= kvkLq(S1): the proof is completed with
sup kv∗k Lq (S1)=1 Z π −π f∗2+ g∗2v∗dσ = Z π −π f∗2+ g∗2 p/2 dσ 2/p .
Appendix C: Some numerical results
To compute the curve α 7→ µa,p(α), we systematically
solve the Euler-Lagrange equation associated with the variation of Qa,p,α, i.e.,
− u00+ a2 u3Rπ −π 1 u2dσ 2 + α u = u p−1 (C1)
where the solution u > 0 is normalized by
µa,p(α) Z π −π updσ 2p−1 = 1 .
This condition a posteriori provides the numerical value of µa,p(α). To impose the boundary conditions u0(0) =
u0(π) = 0, we use a shooting method and solve (C1) on R with the conditions u0(0) = 0 and u(0) = λ > 0. To emphasize the dependence in λ, let us denote it by
uλ. For any λ > 0, λ 6= (a2+ α)1/(p−2), the solution is
non-constant and periodic so that
ρ(λ) = min{s > 0 : u0λ(s) = 0}
is well defined. The shooting parameter λ is then de-termined by the condition that ρ(λ) = π. Since (C1) involves a nonlocal term, an additional fixed-point pro-cedure is needed to adjust the coefficient of u−3 in the equation. Some plots are shown in Figs.1and2.
-0.2 -0.1 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5
FIG. 1. The curve α 7→ µa,p(α) with p = 4 and a = 0.45.
The only solutions to (C1) are the constant functions for any α such that − a2 = − 0.2025 ≤ α ≤ − 0.1075 and, in this range, µa,p(α) = a2+ α. A branch of non-constant optimizers
of (2) bifurcates at α = − 0.1075. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2
FIG. 2. The curve α 7→ µa,p(α) with p = 4 and a = 0.2.
Here the branch of non-constant optimizers of (2) bifurcates at α = 0.38 which corresponds to a2(p + 2) + α (p − 2) = 1.
Equality in (2) is achieved only by constant functions according to LemmaIII.4 if a2(p + 2) + α (p − 2) ≤ 1: in this case, λ = (a2+ α)1/(p−2) ≡ u
λ. For any a ∈
(0, 1/2) such that a2(p + 2) + α (p − 2) > 1, our method
provides us with a non-constant solution u of (C1) which realizes the equality in (2). As a → 1/2, the integral Rπ
−πu−2dσ diverges, so that the limit curve is described
by the solution of
with boundary conditions u0(0) = 0 and u(π) = 0. See Fig.3. 0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.2 0.3 0.4 0.5 0.6
FIG. 3. Here p = 4 and α = 0. Plot of the solution of (C1) for a = 0.40, 0.41,. . . 0.49. The thick curve solves u00+ up−1= 0 and it is explicit. Similar patterns are found when α 6= 0, with a non-explicit curve solving (C2) in the limit as a → 1/2.
AL was partially supported by RSF grant No. 18-11-00032
ACKNOWLEDGMENTS
This research has been partially supported by the project EFI, contract ANR-17-CE40-0030 (J.D.) of the French National Research Agency (ANR), by RSF grant No. 18-11-00032 (A.L.) and by the NSF grant DMS-DMS-1600560 (M.L.). Some of the preliminary investigations were done at the In-stitute Mittag-Leffler during the fall program Interactions be-tween Partial Differential Equations & Functional Inequali-ties. The authors thank A. I. Nazarov for pointing them sev-eral important references.
c
2018 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
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