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Forced vibrations of wave equations with non-monotone nonlinearities
Vibrations forcées d’équations des ondes avec des non-linéarités non-monotones
Massimiliano Berti
a, Luca Biasco
b,∗aSISSA, via Beirut 2-4, Trieste, Italy
bUniversità di Roma3, Largo S. Leonardo Murialdo, Roma, Italy
Received 15 December 2004; received in revised form 11 May 2005; accepted 20 May 2005 Available online 19 January 2006
Abstract
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov–
Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods related to the regularity theory of [P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967) 145–205].
©2005 Elsevier SAS. All rights reserved.
Résumé
On prouve l’existence et la régularité de solutions périodiques en temps d’équations des ondes non linéaires forcées, complè- tement résonnantes, avec des conditions au bord de Dirichlet, pour une grande classe de termes forcants non-monotones. Notre approche est basée sur une réduction de Lyapunov–Schmidt variationnelle. L’équation de bifurcation en dimension infinie présente un manque intrinsèque de compacité. Nous la résolvons par un argument de minimisation et à l’aide d’estimations a priori inspirées de la théorie de la régularité de [P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm.
Pure Appl. Math. 20 (1967) 145–205].
©2005 Elsevier SAS. All rights reserved.
MSC:35L05; 35L20; 35B10; 37K10
Keywords:Wave equation; Periodic solutions; Variational methods; A-priori estimates; Lyapunov–Schmidt reduction
* Corresponding author. Tel.: +39 040 3787449; fax +39 040 3787528.
E-mail addresses:berti@sissa.it (M. Berti), biasco@mat.uniroma3.it (L. Biasco).
0294-1449/$ – see front matter ©2005 Elsevier SAS. All rights reserved.
doi:10.1016/j.anihpc.2005.05.004
1. Introduction
In this paper we consider the problem of finding nontrivial time-periodic solutions of the completely resonant nonlinearforcedwave equation
u=εf (t, x, u;ε) (1.1)
with Dirichlet boundary conditions
u(t,0)=u(t, π )=0 (1.2)
where:=∂t t−∂xxis the D’Alembertian operator,εis a small parameter and the nonlinear forcing termf (t, x, u;ε) isT-periodic in time. We consider the case whenT is a rational multiple of 2πand, for simplicity of exposition, we shall assume
T =2π.
We look for nontrivial 2π-periodic in time solutionsu(t, x)of (1.1), (1.2), i.e. satisfying
u(t+2π, x)=u(t, x). (1.3)
Forε=0, (1.1), (1.2) reduces to the linear homogeneous wave equation u=0,
u(t,0)=u(t, π )=0 (1.4)
which possesses aninfinitedimensional space of solutions which are 2π-periodic in time and of the formv(t, x)= ˆ
v(t+x)− ˆv(t−x) for any 2π-periodic function v(ˆ ·). For this reason equation (1.1), (1.2) is called completely resonant.
The main difficulty for proving existence of solutions of (1.1)–(1.3) forε=0 is to find from which periodic orbits of the linear equation (1.4) the solutions of the nonlinear equation (1.1) branch off. This requires to solve an infinite dimensional bifurcation equation (also called kernel equation) with an intrinsiclack of compactness.
The first breakthrough regarding problem (1.1)–(1.3) was achieved by Rabinowitz in [18] where existence and regularity of solutions was proved for nonlinearities satisfying the strongly monotone assumption(∂uf )(t, x, u) β >0. Using methods inspired by the theory of elliptic regularity, [18] proved the existence of a unique curve of smooth solutions forεsmall. Other existence results of weak and classical solutions have been obtained, still in the strongly monotone case, in [10,12,16].
Subsequently, Rabinowitz [19] was able to prove existence of weak solutions of (1.1)–(1.3) for a class of weakly monotone nonlinearities likef (t, x, u)=u2k+1+G(t, x, u)whereG(t, x, u2)G(t, x, u1)ifu2u1. Actually, in [19] bifurcation of a global continuum branch of weak solutions is proved. For other local existence results in the weakly monotone case we mention [14,20].
In all the quoted papers the monotonicity assumption (strong or weak) is the key property for overcoming the lack of compactness in the infinite dimensional kernel equation.
We underline that, in general, the weak solutions obtained in [19] are only continuous functions. Concerning regularity, Brezis and Nirenberg [10] proved – but only for strongly monotone nonlinearities – that anyL∞-solution of (1.1)–(1.3) is smooth, even in the nonperturbative caseε=1, whenever the nonlinearityf is smooth.
On the other hand, very little is known about existence and regularity of solutions if we drop the monotonicity assumption on the forcing termf. Willem [21], Hofer [15] and Coron [11] have considered the class of equations (1.1), (1.2) wheref (t, x, u)=g(u)+h(t, x),ε=1, andg(u)satisfies suitable linear growth conditions. Existence of weak solutions is proved, in [15,21], for a set ofhdense inL2, although explicit criteria that characterize suchh are not provided. The infinite dimensional bifurcation problem is overcome by assuming non-resonance hypotheses between the asymptotic behavior ofg(u)and the spectrum of . On the other side, Coron [11] finds weak solu- tions assuming the additional symmetryh(t, x)=h(t+π, π−x)and restricting to the space of functions satisfying u(t, x)=u(t+π, π−x), where the Kernel of the d’Alembertian operatorreduces to 0. Actually [11] also deals with the autonomous caseh≡0,proving for the first time existence of nontrivial solutions for non-monotone nonlin- earities. For some more recent results see for example [3].
In the present paper we prove existence and regularity of solutions of (1.1)–(1.3) for a large class ofnon-monotone forcing termsf (t, x, u), including, for example,
f (t, x, u)= ±u2k+h(t, x), see Theorem 1;
f (t, x, u)= ±u2k+u2k+1+h(t, x), see Theorem 2;
f (t, x, u)= ±u2k+ ˜f (t, x, u) with(∂uf )(t, x, u)˜ β >0, see Theorem 3.
The precise results were announced in [4] and will be stated in the next Subsection 1.1, see Theorems 1, 2 and 3.
Their proof is based on a variational Lyapunov–Schmidt reduction, minimization arguments and a-priori estimate methods inspired to regularity theory of [18]. We anticipate that our approach – explained in Subsection 1.2 – is not merely a sharpening of the ideas of [18,19], which, to deal with non-monotone nonlinearities, require a significant change of perspective.
We mention that in the last years several results on bifurcation of free vibrations for completely resonant au- tonomous wave equations have been proved in [2,5–8,13]. The main differences with respect to the present case are that: a “small divisor” problem in solving the “range equation” appears (here no small divisor problem is present due to the assumptionT =2π, see Remark 1.7), but the infinite dimensional “bifurcation equation” – whose solutions is the main problem of the present paper – gains crucial compactness properties, see Remark 1.8.
1.1. Main results
We look for solutionsu:Ω→Rof (1.1) in the Banach space E:=H1(Ω)∩C1/20 Ω
, Ω:=T×(0, π ),
where H1(Ω) is the usual Sobolev space and C01/2(Ω) is the space of all the 1/2-Hölder continuous functions u:Ω→Rsatisfying (1.2), endowed with norm1
u E:= u H1(Ω)+ u C1/2(Ω).
Critical points of the Lagrangian action functionalΨ ∈C1(E,R) Ψ (u):=Ψ (u, ε):=
Ω
u2t 2 −u2x
2 +εF (t, x, u;ε)
dtdx, (1.5)
whereF (t, x, u;ε):=u
0 f (t, x, ξ;ε)dξ, are weak solutions of (1.1)–(1.3).
Forε=0, the critical points ofΨ inEreduce to the solutions of the linear equation (1.4) and form the subspace V :=N∩H1(Ω)where
N:= v(t, x)= ˆv(t+x)− ˆv(t−x)
vˆ∈L2(T)and 2π 0
ˆ
v(s)ds=0
. (1.6)
Note thatV :=N∩H1(Ω)= {v(t, x)= ˆv(t+x)− ˆv(t−x)∈N| ˆv∈H1(T)} ⊂E, since any functionvˆ∈H1(T)is 1/2-Hölder continuous.
LetN⊥:= {h∈L2(Ω)|
Ωhv=0,∀v∈N}denote theL2(Ω)-orthogonal ofN. We prove the following theorem:
Theorem 1.Letf (t, x, u)=βu2k+h(t, x)and h∈N⊥satisfies h(t, x) >0 (or h(t, x) <0)a.e. inΩ. Then, for ε small enough, there exists at least one weak solution u∈E of (1.1)–(1.3)with u EC|ε|. If, moreover,h∈ Hj(Ω)∩Cj−1(Ω), j 1, then u∈Hj+1(Ω)∩C0j(Ω) with u Hj+1(Ω)+ u Cj(Ω) C|ε| and therefore, for j 2, uis a classical solution.
1 Here u2
H1(Ω):= u2
L2(Ω)+ ux 2
L2(Ω)+ ut 2 L2(Ω)and u C1/2(Ω) := u C0(Ω)+ sup
(t,x)=(t1,x1)
|u(t, x)−u(t1, x1)| (|t−t1| + |x−x1|)1/2.
Theorem 1 is a corollary of the following more general result which enables to deal with non-monotone nonlinear- ities like, for example,f (t, x, u)= ±(sinx)u2k+h(t, x),f (t, x, u)= ±u2k+u2k+1+h(t, x).
Theorem 2.Letf (t, x, u)=g(t, x, u)+h(t, x),h(t, x)∈N⊥and g(t, x, u):=β(x)u2k+R(t, x, u)
whereR, ∂tR,∂uR∈C(Ω×R,R)satisfy2 R(·, u)
C(Ω) =o u2k
, ∂tR(·, u)C(
Ω)=O u2k
, ∂uR(·, u)
C(Ω) =o u2k−1
, (1.7)
andβ∈C([0, π],R)verifies, forx∈(0, π ),β(x) >0 (orβ(x) <0)andβ(π−x)=β(x).
(i) (Existence) Assume there exists a weak solutionH∈EofH=hsuch that
H (t, x) >0 (orH (t, x) <0) ∀(t, x)∈Ω. (1.8)
Then, forεsmall enough, there exists at least one weak solutionu∈Eof(1.1)–(1.3)satisfying u EC|ε|.
(ii) (Regularity) If, moreover, h∈Hj(Ω)∩Cj−1(Ω), β∈Hj((0, π )),R,∂tR, ∂uR∈Cj(Ω×R), j 1, then u∈Hj+1(Ω)∩C0j(Ω) and, forj2,uis a classical solution.
Note that Theorem 2 does not require any growth condition ongat infinity. In particular it applies for any analytic functiong(u)satisfyingg(0)=g(0)= · · · =g2k−1(0)=0 andg2k(0)=0.
We now collect some comments on the previous results.
Remark 1.1.The assumptionh∈N⊥ is not of technical nature both in Theorem 1 and in Theorem 2 (at least if g=g(x, u)=g(x,−u)=g(π−x, u)). Indeed, ifh /∈N⊥, periodic solutions of problem (1.1)–(1.3) do not exist in any fixed ball{ u L∞R},R >0, forεsmall; see Remark 4.5.
We also note that the vector space of theh∈L2(Ω)verifyingh(t, x)=h(t+π, π−x), introduced in [11], is a subspace ofN⊥.
Remark 1.2.In Theorem 2 hypotheses (1.8) andβ >0 (orβ <0) are assumed to prove the existence of a minimum of the “reduced action functional”Φ, see (1.16). A sufficient condition implying (1.8) ish >0 a.e. inΩ, see the
“maximum principle” Proposition 4.11. This is also the key step to derive Theorem 1 from Theorem 2.
We also note that hypotheses (1.8) can be weakened, see Remark 4.4.
Remark 1.3((Regularity)).It is quite surprising that the weak solutionuof Theorems 1, 2 is actually smooth. Indeed, while regularity always holds true for strictly monotone nonlinearities (see [10,18]), yet for weakly monotonef it is not proved in general, unless the weak solutionuverifies ΠNu L2 C >0 (see [19]). Note, on the contrary, that the weak solutionuof Theorem 2 satisfies ΠNu L2=O(ε).
Moreover, assuming ∂tl∂xm∂unR
C(Ω) =O u2k−n
, ∀0l, nj+1, 0mj, l+m+nj+1 (1.9)
we can also prove the estimate (see Remark 4.10)
u Hj+1(Ω)+ u Cj(Ω) C|ε|. (1.10)
Remark 1.4((Multiplicity)).For non-monotone nonlinearitiesf one can NOTin general expect uniqueness of the solutions. Actually, for f (t, x, u)=g(x, u)+h(t, x)with g(x, u)=g(x,−u), g(π−x, u)=g(x, u), there exist infinitely manyh∈N⊥for which problem (1.1)–(1.3) has (at least) 3 solutions, see Remark 4.6.
Remark 1.5((Minimal period)).Ifh(t, x)has minimal period 2πw.r.t. time, then also the solutionu(t, x)has minimal period 2π, see Remark 4.7.
2 The notationf (z)=o(zp),p∈N, means thatf (z)/|z|p→0 asz→0.f (z)=O(zp)means that there exists a constantC >0 such that
|f (z)|C|z|pfor allzin a neighborhood of 0.
Finally, we extend the result of [18] proving existence of periodic solutions for non-monotone nonlinearities f (t, x, u)obtained adding to a nonlinearityf (t, x, u)˜ as in [18] (i.e.∂uf˜β >0) any non-monotone terma(x, u) satisfying
a(x,−u)=a(x, u), a(π−x, u)=a(x, u) (1.11)
or
a(x,−u)= −a(x, u), a(π−x, u)= −a(x, u). (1.12)
A prototype nonlinearity isf (t, x, u)= ±u2k+ ˜f (t, x, u)with∂uf˜β >0.
Theorem 3.Letf (t, x, u)= ˜f (t, x, u)+a(x, u)wheref,∂tf,∂uf are continuous,∂uf˜β >0anda(x, u)satisfy (1.11) or(1.12). Then, for εsmall enough, (1.1)–(1.3)has at least one weak solutionu∈E. If moreoverf,∂tf,
∂uf ∈Cj(Ω×R),j1, thenu∈Hj+1(Ω)∩Cj0(Ω).
In the next subsection we describe our method of proof.
1.2. Scheme of the proof
In order to find critical points of the Lagrangian action functionalΨ:E→Rdefined in (1.5) we perform a varia- tional Lyapunov–Schmidt reduction, decomposing the spaceE:=H1(Ω)∩C01/2(Ω)as
E=V⊕W where
V :=N∩H1(Ω) and W:=N⊥∩H1(Ω)∩C01/2Ω .
Setting u=v+w withv∈V,w∈W and denoting byΠN andΠN⊥ the projectors from L2(Ω)ontoN andN⊥ respectively, problem (1.1)–(1.3) is equivalent to solve thekernel equation
ΠNf (v+w, ε)=0 (1.13)
and therange equation
w=ε−1ΠN⊥f (v+w, ε) (1.14)
where−1:N⊥→N⊥is the inverse ofandf (u, ε)denotes the Nemitski operator associated tof, namely f (u, ε)
(t, x):=f (t, x, u, ε).
Remark 1.6. The usual approach (see [12,18–20]) is to find, first, by the monotonicity off, the unique solution v=v(w) of the kernel equation (1.13) and, next, to solve the range equation (1.14). On the other hand, for non- monotone forcing terms, one can notin general solve uniquely the kernel equation – recall by Remark 1.4 that in general uniqueness of solutions does not hold. Therefore we must solve first the range equation and thereafter the kernel equation. For other applications of this approach to perturbation problems in critical point theory, see e.g. the forthcoming monograph by Ambrosetti and Malchiodi [1].
We solve, first, the range equation by means of a quantitative version of the Implicit Function Theorem, finding a solutionw:=w(v, ε)∈W of (1.14) with w(v, ε) E=O(ε), see Proposition 3.2. Here no serious difficulties arise since−1acting onWis a compact operator, due to the assumptionT =2π, see (2.2).
Remark 1.7.More in general,−1is compact on the orthogonal complement of ker()wheneverT is a rational multiple of 2π. On the contrary, ifT is an irrational multiple of 2π, then−1is, in general, unbounded (a “small divisor” problem appears), but the kernel ofreduces to 0 (there is no bifurcation equation). For existence of periodic solutions in the caseT /2πis irrational see [17].
Once the range equation (1.14) has been solved byw(v, ε)∈Wit remains the infinite dimensional kernel equation (also called bifurcation equation)
ΠNf
v+w(v, ε), ε
=0. (1.15)
We note (see Lemma 3.3) that (1.15) is the Euler–Lagrange equation of thereduced Lagrangian action functional Φ:V −→R, Φ(v):=Φ(v, ε):=Ψ
v+w(v, ε), ε
. (1.16)
Φlacks compactness properties and to find critical points ofΦwe cannot rely on critical point theory.
Remark 1.8.Implementing an analogue Lyapunov–Schmidt reduction in the autonomous case (see [5]) it turns out that, in the corresponding reduced Lagrangian action functional, a further term proportional to v 2
H1 is present.
Therefore it is possible to apply critical point theory (e.g. the Mountain Pass Theorem) to find existence and multi- plicity of solutions, see [6]. The elliptic term v 2
H1 helps also in proving regularity results for the solutions.
We attempt to minimizeΦ.
We do not try to apply the direct methods of the calculus of variations. IndeedΦ, even though it could possess some coercivity property, will not be convex (beingf non-monotone). Moreover, without assuming any growth condition on the nonlinearityf, the functionalΦ could neither be well defined on anyLp-space.
Therefore we minimizeΦ in anyBR:= {v∈V , v H1R},∀R >0, as in [18]. By standard compactness argu- mentsΦ attains minimum at, say,v¯∈BR. Since v¯ could belong to the boundary∂BR,v¯could not be a solution of (1.15) and we can only conclude the variational inequality
DvΦ(v)¯ [ϕ] =
Ω
f
¯
v+w(v, ε), ε¯
ϕ0 (1.17)
for anyadmissible variationϕ∈V, i.e. ifv¯+θ ϕ∈BR,∀θ <0 sufficiently small.
The heart of the existence proof of the weak solutionu of Theorems 1, 2 and 3 is to obtain, choosing suitable admissible variations like in [18], the a-priori estimate ¯v H1 < R for someR >0, i.e. to show thatv¯ is aninner minimum point ofΦ inBR.
The strong monotonicity assumption(∂uf )(t, x, u)β >0 would allow here to get such a-priori estimates by arguments similar to [18]. On the contrary, the main difficulty for proving Theorems 1, 2 and 3 which deal with non-monotone nonlinearities is to obtain such a-priori-estimates forv.¯
The most difficult cases are the proof of Theorems 1 and 2. To understand the problem, let consider the particular nonlinearity f (t, x, u)=u2k+h(t, x) of Theorem 1. The even term u2k does not give any contribution into the variational inequality (1.17) at the 0th-order inε, since the right-hand side of (1.17) reduces, forε=0, to
Ω
v¯2k+h(t, x)
ϕ=0, ∀ϕ∈V
by (2.18) andh∈N⊥.
Therefore, for deriving, if ever possible, the required a-priori estimates, we have to develop the variational inequal- ity (1.17) at higher orders inε. We obtain
0
Ω
2kv¯2k−1ϕw(ε,v)¯ +O
w2(ε,v)¯
=
Ω
ε2kv¯2k−1ϕ−1
h+ ¯v2k +O
ε2
(1.18)
becausew(v, ε)¯ =ε−1(v¯2k+h)+o(ε)(recall thatv¯2k,h∈N⊥).
We now sketch how theε-order term in the variational inequality (1.18) allows to prove anL2k-estimate forv.¯ Inserting the admissible variationϕ:= ¯vin (1.18) we get
Ω
Hv¯2k+ ¯v2k−1v¯2kO(ε) (1.19)
whereH is a weak solution of H =hwhich verifiesH (t, x) >0 inΩ (H exists by the “maximum principle”
Proposition 4.11). The crucial fact is that the first term in (1.19) satisfies the coercivity inequality
Ω
H v2kc(H )
Ω
v2k, ∀v∈V (1.20)
for some constantc(H ) >0, see Proposition 4.2. The second term
Ωv¯2k−1v¯2k will be negligible,ε-close to the origin, with respect to
ΩH v2kand (1.19), (1.20) will provide theL2k-estimate forv.¯
We remark that the inequality (1.20) is not trivial becauseH vanishes at the boundary (H (t,0)=H (t, π )=0).
Actually, the proof of (1.20) relies on the formv(t, x)= ˆv(t+x)− ˆv(t−x)of the functions ofV.
Next, we can obtain, choosing further admissible variationsϕ in (1.18) and using inequalities similar to (1.20), an L∞-estimate for v¯ and, finally, the required H1-estimate, proving the existence of a weak solutionu∈E, see Section 4.
Moreover, using similar techniques inspired to regularity theory and further suitable variations, we can also obtain a-priori estimates for theL∞-norm of the higher order derivatives ofv¯and for itsHj-Sobolev norms. In this way we can prove the regularity of the solutionu– fact quite surprising for non-monotone nonlinearities –, see Subsection 4.5.
Theorem 2 is proved developing such ideas and a careful analysis of the further termR.
The proof of Theorem 3 is easier than for Theorems 1 and 2. Indeed the additional terma(x, u)does not contribute into the variational inequality (1.17) at the 0-order inε, because
Ωa(x,v)ϕ¯ ≡0,∀ϕ∈V, by (2.19). Therefore the dominant term in the variational inequality (1.17) is provided by the monotone forcing termf˜and the required a-priori estimates are obtained with arguments similar to [18], see Section 5.
Notations. Ω:=T×(0, π )whereT:=R/2πZ. We denote byCj(Ω) the Banach space of functions u:Ω→R with j derivatives in Ω continuous up to the boundary ∂Ω, endowed with the standard norm · Cj.C0j(Ω):=
Cj(Ω) ∩C0(Ω) whereC0(Ω) is the space of real valued continuous functions satisfyingu(t,0)=u(t, π )=0.
MoreoverHj(Ω):=Wj,2(Ω)are the usual Sobolev spaces with scalar product·,·Hj and norm 2Hj(Ω) . Here Cj(T)denotes the Banach space of periodic functionsu:T→Rwithj continuous derivatives. Finally,Hj(T)is the usual Sobolev space of 2π-periodic functions.
2. Preliminaries
We first collect some important properties on the D’Alembertian operator.
Definition 2.1.Givenf (t, x)∈L2(Ω), a functionu∈L2(Ω)is said to be a weak solution ofu=f inΩ satisfying the boundary conditionsu(t,0)=u(t, π )=0, iff
Ω
uϕ=
Ω
f ϕ ∀ϕ∈C02Ω .
It is easily verified that, ifu∈C2(Ω) is a weak solution ofu=f according to Definition 2.1, thenuis a classical solution andu(t,0)=u(t, π )=0.
The kernel N⊂L2(Ω)of the D’Alembertian operator, i.e. the space of weak solutions of the homogeneous linear equationv=0 verifying the Dirichlet boundary conditionsv(t,0)=v(t, π )=0, is the subspaceN defined in (1.6). N coincides with the closure inL2(Ω) of the classical solution of v=0 verifying Dirichlet boundary conditions which, as well known, are of the formv(t, x)= ˆv(t+x)− ˆv(t−x),vˆ∈C2(T).
Using Fourier series we can also characterizeNas N=
v(t, x)∈L2(Ω)v(t, x)=
j∈Z
ajeij tsinj xwith
j∈Z
|aj|2<∞
.
The range ofinL2(Ω)is
N⊥:=
f ∈L2(Ω)
Ω
f v=0,∀v∈N
=
f (t, x)∈L2(Ω)f (t, x)=
l∈Z, j1 j=|l|
fljeiltsinj xwith
l∈Z, j1 j=|l|
|flj|2<∞
,
i.e.∀f (t, x)∈N⊥there exists a unique weak solutionu=−1f ∈N⊥ofu=f. Furthermore−1is a bounded operator such that
−1:N⊥−→N⊥∩H1(Ω)∩C01/2Ω
, (2.1)
i.e. there exists a suitable constantc1 such that −1f
Ec f L2 ∀f ∈L2(Ω) (2.2)
where u E:= u H1+ u C1/2. By (2.2) and the compact embeddingH1(Ω) →L2(Ω), the operator−1:N⊥→ N⊥is compact.
These assertions follow easily from the Fourier series representation (see e.g. [10]) f (t, x):=
j1, j=|l|
fljeiltsinj x ⇒ u=−1f:=
j1, j=|l|
flj
−l2+j2eiltsinj x
noting thatuis a weak solution of (1.1) (according to Definition 2.1) iffulj =flj/(−l2+j2)∀l∈Z,j1, see e.g.
[12,15].
To continue,−1is a bounded operator also between the spaces L∞(Ω)−→C0,1Ω
, Hk(Ω)−→Hk+1(Ω), CkΩ
−→Ck+1Ω
(2.3) as follows by the integral formula foru=−1f =ΠN⊥ψwhere (see e.g. [9,16])
ψ (t, x):= −1 2 π x
t−x+ξ t+x−ξ
f (ξ, τ )dτdξ+cπ−x π , ΠN⊥:L2(Ω)→N⊥is the orthogonal projector ontoN⊥and
c:=1 2
π 0
t+ξ
t−ξ
f (ξ, τ )dτdξ≡1 2
π 0
ξ
−ξ
f (ξ, τ )dτdξ≡const (2.4)
is a constant independent oft, because3f ∈N⊥.
3 We have that 2c=
T(t )f=limn→∞
T(t )fnwhereT(t):= {(τ, ξ )∈Ωs.t.t−ξ < τ < t+ξ, 0< ξ < π}and fn(t, x):=
|l|, jn j=|l|
fl,jeiltsinj x L
−→2 j=|l|
fl,jeiltsinj x=f (t, x)∈N⊥.
The claim follows since
T(t )fnis, for anyn, independent ont:
T(t ) fn=
π
0
|l|, jn j=|l|
fl,jsinj ξ t+ξ t−ξ
eiltdτdξ
=
1jn
f0,j π
0
2ξsinj ξdξ+
|l|, jn j=|l|, l=0
2fl,j l eilt
π
0
sinj ξsinlξdξ= 1jn
f0,j π
0
2ξsinj ξdξ.
We also have, since∂tjH is a weak solution of(∂tjH )=∂tjhand (2.1) applies, h∈Hj(Ω) ⇒ ∂tjH∈C1/20 Ω
. (2.5)
Finally, the projectorΠN:L2(Ω)→N can be written asΠNu=p(t+x)−p(t−x)where p(y):= 1
2π π 0
u(y−s, s)−u(y+s, s) ds
and therefore, sinceu∈Cj(Ω) ⇒p∈Cj(T)andu∈Hj(Ω)⇒p∈Hj(T), ΠN, ΠN⊥:Cj(Ω) →CjΩ
are bounded, (2.6)
ΠN, ΠN⊥:Hj(Ω)→Hj(Ω) are bounded. (2.7)
2.1. Kernel properties and technical lemmata Let define, for 0α <1/2,
Ωα:=T×(απ, π−απ )⊆Ω. (2.8)
Lemma 2.2.Leta∈L1(Ω).
Ωα
a(t, x)dtdx=1 2
2π 0
s+−2απ
−2π+s++2απ
a
s++s−
2 ,s+−s− 2
ds−ds+. (2.9)
In particular forp, q∈L1(T)
Ωα
p(t+x)q(t−x)dtdx=1 2
2π 0
p(s)ds 2π 0
q(s)ds−1 2
2απ
−2απ
2π 0
p(y)q(z+y)dydz (2.10)
and
Ωα
p(t+x)dtdx=
Ωα
p(t−x)dtdx=π(1−2α) 2π
0
p(s)ds. (2.11)
Moreover, givenf, g:R→Rcontinuous,
Ωα
f
p(t+x) g
p(t−x) dtdx=
Ωα
f
p(t−x) g
p(t+x)
dtdx. (2.12)
Proof. In Appendix A. 2
Lemma 2.3.For anyv= ˆv(t+x)− ˆv(t−x):=v+−v−∈N
v 2L2(Ω)=2π ˆv 2L2(T)=2π 2π 0
ˆ
v2. (2.13)
Moreover vt 2
L2(Ω)= vx 2
L2(Ω)=2π ˆv 2L2(T) ∀v∈N∩H1(Ω), (2.14)
ˆv L∞(T) v L∞(Ω)2 ˆv L∞(T) ∀v∈N∩L∞(Ω), (2.15)
v L∞(Ω) v H1(Ω) ∀v∈N∩H1(Ω). (2.16)
Proof. In Appendix A. 2
Lemma 2.4.Letϕ1, . . . , ϕ2k+1∈Nandϕ1· · ·ϕ2k+1∈L1(Ω). Then
Ωα
ϕ1· · ·ϕ2k+1=0. (2.17)
In particularϕ1· · ·ϕ2k∈N⊥.
Moreover, ifa:Ω→Rsatisfies(i)a(x, u)=a(π−x, u)anda(x, u)=a(x,−u)or(ii)a(x, u)= −a(π−x, u) anda(x, u)= −a(x,−u), then
Ωα
a(x, v)ϕ=0 ∀v∈N∩L∞, ϕ∈N (2.18)
and
Ωα
(∂ua)(x, v)ϕ1ϕ2=0 ∀v∈N∩L∞, ϕ1, ϕ2∈N. (2.19)
Proof. In Appendix A. 2
Lemma 2.5.The following inequalities hold:
(a−b)2k22k−1
a2k+b2k
∀a, b∈R, (2.20)
(a−b)2ka2k+b2k−2k
a2k−1b+ab2k−1
∀a, b∈R, k∈N, k2, (2.21)
(a+b)2k−1−a2k−141−kb2k−1 ∀a∈R, b >0, k∈N+. (2.22) Proof. In Appendix A. 2
2.2. Generalities about the difference quotients
Forf ∈L2(Ω)we define the difference quotient of sizeh∈R\ {0} (Dhf )(t, x):=f (t+h, x)−f (t, x)
h and theh-translation
(Thf )(t, x):=f (t+h, x) with respect to time.
The following lemma collects some elementary properties of the difference quotient.
Lemma 2.6.Letf, g∈L2(Ω),h∈R\ {0}. The following holds (i) Leibnitz rule:
Dh(f g)=(Dhf )g+Thf Dhg, (2.23)
Dhfm=(Dhf )
m−1 j=0
fm−j−1Thfj=m(Dhf )fm−1+(Dhf )
m−1 j=0
fm−j−1
Thfj−fj
, (2.24)
Ω
Dh(f g)=
Ω
(Dhf )g+f (D−hg); (2.25)
(ii) integration by parts:
Ω
f (D−hg)= −
Ω
(Dhf )g; (2.26)
(iii) weak derivative:If there exists a constantC >0such that∀hsmall
Dhf L2 C ⇒ thenf has a weak derivativeftand ft L2C. (2.27) Moreover, iff has a weak derivativeft∈L2(Ω), then
(iv) estimate on the difference quotient:
Dhf L2(Ω) ft L2(Ω); (2.28)
(v) convergence:
Dhf L
−→2 ft ash−→0. (2.29)
Proof. In Appendix A. 2
3. The Lyapunov–Schmidt decomposition 3.1. The range equation
We first solve the range equation (1.14) applying the following quantitative version of the Implicit Function Theo- rem, whose standard proof is omitted.
Proposition 3.1. Let X, Y, Z be Banach spaces and x0∈X, y0∈Y. Fix r, ρ >0 and define Xr := {x ∈Xs.t.
x−x0 X< r}andYρ:= {y∈Ys.t. y−y0 Y< ρ}. LetF∈C1(U, Z)whereYρ×Xr ⊂U⊂Y×Xis an open set.
Suppose that
F(y0, x0)=0 (3.1)
and thatDyF(y0, x0)∈L(Y, Z)is invertible. LetT :=(DyF(y0, x0))−1and T := T L(Z,Y )be its norm. If sup
Xr
F(y0, x)
Z ρ
2 T , (3.2)
sup
Yρ×Xr
IdY−T DyF(y, x)
L(Y,Y )1
2, (3.3)
then there existsy∈C1(Xr, Yρ)such thatF(y(x), x)≡0.
Applying Proposition 3.1 to the range equation (1.14) we derive:
Proposition 3.2.Letf =f (t, x, u, ε),fu:=∂uf andε∂εf be continuous onΩ×R× [−1,1]. Then∀R >0there exists a unique function
w=w(v, ε)∈C1
v L∞<2R
×
|ε|< ε0(R) ,
w EC0(R)|ε|
(3.4) solving the range equation(1.14), whereε0(R):=1/2C0(R)and4
C0(R):=1+√
2π c max
Ω×{|u|3R+1}×{|ε|1}f (t, x, u, ε)+fu(t, x, u, ε). (3.5)
4 cis defined in (2.2).
Proof. LetX:=V ×R, Y =Z:=W(namelyx:=(v, ε)andy:=w) and x X:= v L∞+ε0R(R)|ε|. Let alsox0:=
(0,0),y0:=0,r:=3R,ρ:=1,F(y, x):=F(w, v, ε):=w−ε−1ΠN⊥f (v+w, ε)andU:=W×V ×(−1,1).
Note thatF(·,·)∈C1since the Nemitski operatorεf ∈C1(E×(−1,1), L2(Ω))and (2.1) holds. Moreover (3.1) holds andT =IdW(hence T =1).
If v L∞3Rand w L∞ w E1, then
v(t, x)+w(t, x)3R+1, ∀(t, x)∈Ω. (3.6)
Using (2.2), Bessel inequality ΠN⊥f L2 f L2, f L2(Ω)√
2π f L∞(Ω) and estimate (3.6), we obtain,
∀|ε|ε0(R)and v L∞3R, F(0, v, ε)
E|ε|cf (v, ε)
L2|ε|√2π cf (v, ε)
L∞C0(R)|ε|1
2 (3.7)
whereC0(R)is defined in (3.5). Hence (3.2) follows from (3.7).
Since
DwF(w, v, ε)[w] =w−ε−1ΠN⊥
fu(v+w, ε)w
∀w∈W, we deduce, arguing as before,∀ v L∞3R,∀ w E1,∀|ε|ε0(R),
sup
w E=1w−DwF(w, v, ε)[w]
E|ε|√2π cfu(v+w, ε)
L∞ε0(R)C0(R)=1
2 (3.8)
and (3.3) follows. Now we can apply Proposition 3.1 finding a functionw=w(v, ε)∈C1({ x X< r}, W1)satisfying the range equation (1.14). Finally, note that
v L∞<2R
×
|ε|< ε0(R)
⊂
x X< r=3R and, arguing as above,
w(v, ε)
E=ε−1ΠN⊥f
v+w(v, ε);ε
E|ε|√2π cf (v+w;ε)
L∞|ε|C0(R), whence (3.4) follows. 2
3.2. The kernel equation
Once the range equation (1.14) has been solved byw(v, ε)∈W there remains the infinite dimensional kernel equation (1.15).
SinceV is dense inNwith theL2-norm, Eq. (1.15) is equivalent to
Ω
f
v+w(v, ε), ε
ϕ=0 ∀ϕ∈V (3.9)
which is the Euler–Lagrange equation of thereduced Lagrangian action functionalΦ:V →R,Φ(v):=Φ(v, ε):=
Ψ (v+w(v, ε), ε), defined in (1.16). Actually:
Lemma 3.3.Φ∈C1({ v H1<2R},R)and a critical pointv¯ofΦis a weak solution of the kernel equation(1.15).
MoreoverΦcan be written as Φ(v)=ε
Ω
F
v+w(v);ε
−1 2f
v+w(v);ε w(v)
dtdx (3.10)
and
vn H1, ¯v H1R, vn−→ ¯L∞ v ⇒ Φ(vn)−→Φ(v).¯ (3.11)
Proof. SinceΨ (·, ε)∈C1(E,R)and, by Proposition 3.2,w(·, ε)∈C1({ v H1<2R},R)(note that{ v H1<2R} ⊂ { v L∞<2R}by (2.16)), thenΦ∈C1({ v H1<2R},R)and
DΦ(v)[ϕ] =DΨ
v+w(v)
ϕ+dw(v)[ϕ]
∀ϕ∈V . (3.12)
We claim that, sincew=w(v)∈Eis a weak solution of the range equation (1.14) andw:=dw(v)[ϕ] ∈W, then DΨ
v+w(v)
dw(v)[ϕ]
=0. (3.13)
Indeed, sincevt, vx∈N andwt,wx∈N⊥, DΨ (v+w)[w] =
Ω
(v+w)twt−(v+w)xwx+εf (v+w, ε)w
=
Ω
wtwt−wxwx+εΠN⊥f (v+w, ε)w=0 (3.14) becausew∈Eis a weak solution of the range equationw=εΠN⊥f (v+w, ε)andw(t,0)=w(t, π )=0.
By (3.12), (3.13) and sincewt, wx∈N⊥andϕt, ϕx∈N DΦ(v)[ϕ] =DΨ (v+w)[ϕ] =
Ω
(v+w)tϕt−(v+w)xϕx+εf (v+w, ε)ϕ
=
Ω
vtϕt−vxϕx+εf (v+w, ε)ϕ=ε
Ω
f (v+w, ε)ϕ=ε
Ω
ΠN⊥f (v+w, ε)ϕ (3.15) where in (3.15) we used
Ωvtϕt−vxϕx=0 sincev, ϕ∈V.
Now we prove (3.10) as in [5]. Sincevt, vx∈N,wt, wx∈N⊥and (2.14) Φ(v)=
Ω
(v+w(v))2t
2 −(v+w(v))2x
2 +εF
v+w(v);ε
=
Ω
(w(v))2t
2 −(w(v))2x
2 +εF
v+w(v);ε
and since
Ω(w(v))2t −(w(v))2x= −
Ωεf (v+w(v);ε)w(v)we deduce (3.10).
Finally let us prove (3.11). Settingwn:=w(vn, ε)andw:=w(v, ε), we have¯
Ω
F (vn+wn)−F (v¯+ w )
max
Ω×{|u|R+1}×{|ε|1}
f (t, x, u, ε)
Ω
|vn− ¯v+wn− w| C0(R)
vn− ¯v L1+ wn− w L1
−→0
asn→ ∞, by the fact that vn− ¯v L∞→0 and (3.4). An analogous estimate holds for the second term in the integral in (3.10). 2
By standard compactness argument the functional Φ attains minimum (resp. maximum) in BR := {v ∈ V , v H1 R},∀R >0. Indeed, letvn∈BR be a minimizing (resp. maximizing) sequenceΦ(vn)→infB
RΦ. Since {vn}n∈N is bounded in N∩H1, up to a subsequencevnH
1
v¯ for somev¯= ¯v(R, ε)∈BR. Moreover by the com- pact embeddingH1(T) →L∞(T)we can also assumevn−→ ¯L∞ v(sincevˆnL−→ ˆ¯∞(T)v) and therefore, by (3.11),v¯ is a minimum (resp. maximum) point ofΦrestricted toBR.
Sincev¯ could belong to the boundary∂BRwe only have the variational inequality (1.17) for anyadmissible vari- ationϕ∈V, namely for anyϕ∈V such thatv¯+θ ϕ∈BR,∀θ <0 sufficiently small. As proved by Rabinowitz [18], a sufficient condition forϕ∈V to be an admissible variation is the positivity of the scalar product
¯v, ϕH1>0. (3.16)
The heart of the existence proof of Theorems 1, 2 and 3 is to obtain, choosing suitable admissible variations, the a-priori estimate ¯v H1 < Rfor someR >0, i.e. to show thatv¯ is aninnerminimum (resp. maximum) point ofΦ inBR.
4. Proof of Theorems 1 and 2
The main difficulty for proving Theorems 1 and 2 is to obtain the fore mentioned a-priori-estimate forv.¯
