On some results of Moser and of Bangert

www.elsevier.com/locate/anihpc

On some results of Moser and of Bangert Sur quelques résultats de Moser et Bangert

P.H. Rabinowitz

, E. Stredulinsky

aDepartment of Mathematics University of Wisconsin-Madison, Madison, WI 53706, USA bDepartment of Mathematics University of Wisconsin-Richland, Richland Center, WI 53581, USA

Received 5 May 2003; accepted 17 October 2003 Available online 13 February 2004

Abstract

A new proof is given of results in (V. Bangert, AIHP Anal. Nonlin. 6 (1989) 95) on the existence of minimal (in the sense of Giaquinta and Guisti) heteroclinic solutions of a nonlinear elliptic PDE. Bangert’s work is based on an earlier paper of Moser (AIHP Anal. Nonlin. 3 (1986) 229). Unlike (V. Bangert, AIHP Anal. Nonlin. 6 (1989) 95), the proof here is variational in nature, and involves the minimization of a ‘renormalized’ functional. It is meant to be the first step towards finding locally vs. globally minimal solutions of the PDE.

2004 Elsevier SAS. All rights reserved.

Résumé

Nous donnons une démonstration nouvelle des résultats de Bangert sur l’existence d’une solution minimale (au sens de Giaquinta et Guisti) hétéroclinique d’un EDP elliptique nonlinéaire. Le travail de Bangert est basé sur un article de Moser (AIHP Anal. Nonlin. 3 (1986) 229). Contrairement à (V. Bangert, AIHP Anal. Nonlin. 6 (1989) 95), la démonstration est variationelle en nature, et utilise la minimisation d’une fonctionelle « renormalisée ». C’est une tentative de premier pas pour trouver des solutions minimales localement, plutôt que globalment de l’EDP.

2004 Elsevier SAS. All rights reserved.

MSC: 35J20; 58E15

Keywords: Monotone twist maps; Minimal heteroclinic; Aubry–Mather theory; Renormalized functional

* Corresponding author.

E-mail addresses: rabinowi@math.wisc.edu (P.H. Rabinowitz), estredul@richland.uwc.edu (E. Stredulinsky).

1 This research was sponsored in part by the National Science Foundation under grant #MCS-8110556.

0294-1449/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2003.10.002

1. Introduction

Motivated by work of Aubry [3] and of Mather [7] on monotone twist maps, Moser [8] made the first steps towards finding analogues of their theory in the setting of quasilinear elliptic partial differential equations onRⁿ. He studied the equation

n

i=1

∂

∂xiFpi(x, u, Du)−Fu(x, u, Du)=0, (1.1)

which arises formally as the Euler equation of the functional

Rⁿ

F(x, u, Du)dx, (1.2)

whereF(x, u, p) is e.g. C³ in its arguments and 1-periodic in the components of x∈Rⁿ and inu. F further satisfies structural conditions that imply weak solutions of (1.1) are actually classical solutions of the equation [8].

LetNα denote the set of solutions of (1.1) that have a prescribedα∈Rⁿas rotation vector, that are minimal in the sense of Giaquinta and Guisti [6], and whose graphs viewed onTⁿ⁺¹have no self intersections. The notion of rotation vector here is the extension toRⁿ of the usual rotation number used in dynamical systems. Likewise the minimal and non-self intersection properties are the analogues for solutions of (1.1) of properties of monotone twist maps. Among other things, Moser showedNα= ∅for allα∈Rⁿand obtained various qualitative and quantitative properties for the members ofNα. E.g. forN0, he obtained solutions that are 1-periodic inx₁, . . . , xn. Letting Mα denote the subset ofNα whose existence Moser established, he further provedMα is an ordered set, i.e.

v, w∈Mαimpliesv≡w,v < w, orv > w.

Bangert carried Moser’s analysis further in various ways. Among other things, he showed that wheneverα∈Qⁿ andMαpossesses a gap, i.e. there are adjacentv₀< w₀inMα, then there is aU₁∈Nαwhich is heteroclinic in an appropriate sense fromv₀tow₀and likewise another solution of (1.1) heteroclinic fromw₀tov₀. E.g. ifα=0 so v₀andw₀are adjacent members ofM0, there is aU₁∈N0heteroclinic inx₁fromv₀tow₀. (Similarly there are members ofN0heteroclinic inx_i fromv₀tow₀, 2in.) Moreover ifM¹₀denotes the set of such heteroclinic solutions,M¹₀ is ordered. If there is a gapv₁< w₁inM¹₀, there is aU₂∈M¹₀ which is also heteroclinic inx₂ fromv₁tow₁. Further gap conditions lead to yet more complicated heteroclinics.

There are analogues of U_i in the Aubry–Mather Theory: gaps between periodic invariant sets lead to the existence of heteroclinic invariant curves joining the periodic sets. Moreover given any formal chain of such heteroclinic invariant curves, there are actual invariant curves shadowing the chain. Thus it is natural to seek such shadowing solutions of (1.1). In recent years, variational methods have been devised to carry out such constructions in dynamical systems or PDE settings, see e.g. [10,5,1,2,9]. These methods require a variational characterization of the basic solutions such asU1above. However Bangert’s clever existence argument in [4] is not variational in nature. Therefore as a first step towards constructing more complex solutions of (1.1), in this paper we provide a variational approach to find the type of solutions Bangert discovered. This will only be done for the special but still significant setting ofα=0 andF(x, u, p)=¹₂|p|²+F (x, u)where

(F1) F ∈C²(Rⁿ×R,R),

(F2) F (x, u)is 1-periodic inx1, . . . , xnandu, and (F₃) F is even inx₁, . . . , x_n.

Thus (1.1) becomes

−u+Fu(x, u)=0. (PDE)

A serious difficulty that has to be overcome in the variational approach to (PDE) is that the associated functionals defined on the classes of functions of the heteroclinic types ofU₁,U₂, etc. are infinite. Thus to obtainU₁requires a renormalization, i.e. subtracting an infinite term from the natural functional. Likewise obtainingU₂ requires a second renormalization, etc.

The existence ofU₁will be carried out in Section 2 and its (Giaquinta–Guisti) minimality will be established in Section 3. An inductive argument will then be given in Section 4 to treat the general case.

2. The simplest heteroclinics

In this section, it will be shown how to obtain solutions of (PDE) that are heteroclinic inx1fromv0tow0where v0< w0are an adjacent pair of solutions of (PDE) that are 1-periodic inx1, . . . , xn.

For ease of notation, set L(u)=1

2|∇u|²+F (x, u) withF satisfying (F₁)–(F₃). Let

Γ₀=

u∈W_loc^1,2(Rⁿ)|uis 1-periodic inx₁, . . . , x_n .

For=(₁, . . . , _n)∈Zⁿ, setT ()= [₁, ₁+1] × · · · × [_n, _n+1]. Define J₀(u)=

T (0)

L(u)dx.

Finally define c0= inf

u∈Γ₀J0(u). (2.1)

In [3], Moser showed:

Proposition 2.2. IfM0≡ {u∈Γ₀|J₀(u)=c₀}, then

1^o M0=φand ifu∈M0,uis a classical solution of (PDE).

2^o M0is an ordered set.

A further property ofM0is:

Corollary 2.3. Ifu∈M0,uis even inx₁, . . . , x_n.

Proof. Setζi(x)=(x1, . . . , xi−1,−xi, xi+1, . . . , xn), 1in. Ifu∈M0, thenui(x)=u(ζi(x))∈M0via (F3).

Ifui≡u, by 2^oof Proposition 2.2, either (i)ui(x) > u(x)or (ii)ui(x) < u(x)for allx∈Rⁿ. If (i) occurs, u_i

ζ_i(x)

=u(x) < u_i(x)=u ζ_i(x)

, (2.4)

a contradiction. Similarly (ii) cannot hold. Thusui≡u, 1in. 2 To continue, assume there is a gap inM0:

(∗)0 There are adjacentv0, w0∈M0withv0< w0.

Set Γ1=

u∈W_loc^1,2(Rⁿ)|v0uw0 a.e. anduis 1-periodic inx2, . . . , xn

.

Now the renormalized functionalJ1(u)for this setting can be introduced. Forj∈Zandk∈ {1, . . . , n}, set τ₋^k_ju(x)=u(x1, . . . , xk+j, . . . , xn)

and fori∈Z, define J_1,i(u)=J₀

τ₋¹_iu

−c₀. (2.5)

Alternatively sete₁=(1,0, . . . ,0), . . . , e_n=(0, . . . ,0,1), the usual basis inRⁿ. Then J_1,i(u)=

T (ie₁)

L(u)dx−c₀. Now define

J₁(u)=

i∈Z

J1,i(u).

The next proposition provides the properties ofJ₁that will be required here. Hypothesis (F₃) plays its main role in 1^o–2^o.

Proposition 2.6. Foru∈Γ₁, 1^o J_1,i(u)0 for alli∈Z. 2^o J₁(u)0.

3^o

T (ie1)L(u)dxJ₁(u)+c₀for anyi∈Z.

4^o J₁is weakly lower semicontinuous (lsc) (with respect toW_loc^1,2(Rⁿ)) onΓ₁. Proof. Foru∈Γ1andx1∈ [i+¹₂, i+1], set

ϕ_i⁺(u)=τ₋¹_iu (2.7)

and forx1∈ [i, i+¹₂], set

ϕ_i⁻(u)=τ₋¹_iu. (2.8)

Extendϕ_i^±(u)first as even functions aboutx1=i+¹₂ and then 1-periodically inx1. Continuing to denote these extensions byϕ^±_i (u), their definition impliesϕ_i^±(u)∈Γ0and

J1,i(u)=1 2

J1,i

ϕ⁺_i (u) +J1,i

ϕ⁻_i (u)

0. (2.9)

Thus 1^oand hence 2^ohold. By 1^o, J1,i(u)=

T (ie₁)

L(u)dx−c0J1(u) (2.10)

so 3^ois valid. To prove 4^o, note first thatJ₀is weakly lsc onΓ₀. Let(u_k)be a sequence inΓ₁andu_k→uweakly inW_loc^1,2(Rⁿ)ask→ ∞. Set

J1;p,q(u)≡ q

p

J1,i(u).

Then by the weak lsc ofJ₀, (2.5), (2.9) and 1^o, J_1;p,q(u)=1

2 q

p

J_1,i ϕ⁺_i (u)

+J_1,i ϕ_i⁻(u)

1

2 q

p

lim

k→∞J1,i

ϕ_i⁺(uk) + lim

k→∞J1,i

ϕ⁻_i (uk)

1

2 lim

k→∞

q

p

J_1,i

ϕ_i⁺(u_k) +J_1,i

ϕ⁻_i (u_k)

= lim

k→∞J1;p,q(uk) lim

k→∞J1(uk). (2.11)

Since (2.11) is valid for allpq∈Z, J₁(u) lim

k→∞

J₁(u_k) (2.12)

and 4^oholds. 2

Remark 2.13. The argument used to prove 1^oshows c₀= inf

u∈W^1,2(T (0))

J₀(u)

and ifu∈W^1,2(T (0))withJ (u)=c₀, thenu∈M0. See e.g. [9] for a similar argument.

Next the class of functions that will be used to find a solution of (PDE) heteroclinic inx1fromv0tow0can be introduced. Set

Γ₁=

u∈Γ₁|uτ₋¹₁uandv₀≡u≡w₀ .

The members ofΓ₁automatically are heteroclinic inx₁fromv₀tow₀(in a weak sense) as the next result shows.

Proposition 2.14. Ifu∈Γ₁andJ₁(u) <∞, thenτ_j¹u→v₀andτ₋¹_ju→w₀weakly inW^1,2(T (0))asj→ ∞. Proof. Sincev0τ_j¹(u)w0for allj∈Z, 3^oof Proposition 2.6 shows(τ_j¹u)is bounded inW^1,2(T (0)). This with the monotonicity property,uτ₋¹₁u, shows there is a uniquev∈W^1,2(T (0))such thatτ_j¹u→vweakly in W^1,2(T (0))and strongly inL²(T (0))asj→ ∞and

v₀vuw₀. (2.15)

Since J1(u) <∞, J0(τ_j¹u)→c0 as |j| → ∞. But J0 is weakly lsc so J0(v)c0. Now Remark 2.13 shows J0(v)=c0andv∈M0. Hence (2.15),(∗)0, andu≡w0implyv=v0. Similarlyτ_j¹u→w0asj→ −∞weakly inW^1,2(T (0)). 2

Remark 2.16. In fact, by arguments in [9], the convergence ofτ_j¹uis inW^1,2(T (0)).

Now to obtain a solution of (PDE) heteroclinic inx₁fromv₀tow₀, set c₁= inf

u∈Γ1

J₁(u). (2.17)

Theorem 2.18. LetF satisfy (F₁)–(F₃) and let(∗)₀hold. Then 1^o There is aU1∈Γ1such thatJ1(U )=c1.

2^o Any suchU1is a classical solution of (PDE).

3^o U₁is heteroclinic (inC²) fromv₀tow₀, i.e. U₁−v_{0 C}2(T (j e1))→0 asj→ −∞and U₁−w_{0 C}2(T (j e1))→ 0 asj→ ∞.

4^o v₀< U₁< τ₋¹₁U₁< w₀.

5^o M1≡ {u∈Γ₁|J₁(u)=c₁}is an ordered set.

6^o Ifu∈M1,uis even inx₂, . . . , x_n.

Remark 2.19. In Section 3, it will be further shown thatU1is minimal in the sense of Giaquinta and Guisti.

Proof of Theorem 2.18. Let(uk)⊂Γ₁be a minimizing sequence for (2.17). Normalize(uk)so that fori <0,

T (ie₁)

ukdx1 2

T (0)

(v0+w0)dx <

T (0)

ukdx. (2.20)

That this can be done follows from Proposition 2.14. By 3^o of Proposition 2.6,(uk)is bounded in W_loc^1,2(Rⁿ).

Therefore there is aU₁∈W_loc^1,2(Rⁿ), 1-periodic inx₂, . . . , x_k such that along a subsequence,u_k→U₁weakly in W_loc^1,2(Rⁿ)and strongly inL²_loc(Rⁿ). Hence by the properties ofu_k,v₀U₁τ₋₁U₁w₀and by (2.20),U₁∈Γ₁. Thus

J1(U1)c1. (2.21)

On the other hand, 4^oof Proposition 2.6 andU₁∈Γ₁imply

J1(U1)c1. (2.22)

Combining (2.21)–(2.22) gives 1^oof Theorem 2.18.

To verify 2^o–5^o, modifications of arguments from [9] will be employed. To prove thatU₁satisfies (PDE), let z∈Rⁿ and let B_r(z) be a ball of radiusr aboutz with r < ¹₂. For j ∈Zⁿ, set z_j =z+j. LetU^∗=U₁ in Rⁿ\ _j∈ZⁿBr(zj)andU^∗=u^∗_j inBr(zj)whereu^∗_j minimizes

Br(z_j)

L(ϕ)dx (2.23)

over

Sr(z_j)=

ϕ∈W_loc^1,2(Rⁿ)|ϕ=U₁inRⁿ\B_r(z_j)

. (2.24)

As in Lemma 2.4 of [9], there is auj∈Sr(zj)minimizing (2.23) and any such minimizer of (2.23) is a solution of (PDE) inBr(zj). A priori, there may not be a unique minimizer but as in Lemma 2.5 of [9], the set of minimizers is ordered and there is a unique smallest one which is chosen to beu^∗_j. Note that

Br(zj)

L(U₁)dx

Br(zj)

L(u^∗_j)dx (2.25)

for allj∈Zⁿ. Hence

J₁(U₁)J₁(U^∗). (2.26)

We claimU^∗∈Γ₁and therefore

J1(U^∗)J1(U1). (2.27)

But then by (2.25), for allj∈Zⁿ,

Br(z_j)

L(U₁)dx=

Br(z_j)

L(u^∗_j)dx.

ThusU₁is a minimizer of (2.23) inSr(z_j)and hence a solution of (PDE).

To verify thatU^∗∈Γ₁, observe first that the 1-periodicity ofU₁inx₂, . . . , x_nimplies the same forU^∗. Thus U^∗∈Γ₁if (a)v₀U^∗w₀, (b)U^∗τ₋¹₁U^∗, and (c)v₀≡U^∗≡w₀. To prove (a), sincev₀U₁w₀, it suffices to showv₀u_jw₀inB_r(z_j). Suppose e.g.u_j(y) < v₀(y)for somey∈B_r(z_j). Set ψ_j =min(v₀, u_j). Then ψj =v0inRⁿ\Br(zj). LetSbe a unitn-cube centered atzj. Thenψj=v0inS\Br(zj),ψj =uj neary, and ψj|S extends naturally toRⁿas an element ofΓ0. Continuing to denote this extension byψj,

c₀=J₀(v₀)J₀(ψ_j)

=

S\Br(z_j)

L(v₀)dx+

Br(z_j)∩{v₀u_j}

L(v₀)dx+

Br(z_j)∩{v₀>u_j}

L(u_j)dx. (2.28)

Hence

Br(zj)∩{v0>uj}

L(v0)dx

Br(zj)∩{v0>uj}

L(uj)dx. (2.29)

Setχ_j=max(v₀, u_j)soχ_j∈Sr(z_j)and

Br(zj)

L(u_j)dx

Br(zj)

L(χ_j)dx=

Br(zj)∩{v₀>uj}

L(v₀)dx+

Br(zj)∩{v₀uj}

L(u_j)dx

so

Br(zj)∩{v0>uj}

L(uj)dx

Br(zj)∩{v0>uj}

L(v0)dx. (2.30)

Combining (2.29) and (2.30) yields equality in these expressions and returning to (2.28) shows

c0=J0(v0)J0(ψj)=J0(v0). (2.31)

Henceψj∈M0. But by Proposition 2.2,ψj cannot both=v0inS\Br(zj)and=ujneary. Thus the assumption thatuj(y) < v0(y)is not tenable andv0uj. Similarlyujw0and (a) has been proved.

To obtain (b), suppose not. Then as in the proof of Proposition 2.3 of [9], for somej, there is anx0∈Br(zj) such that

u^∗_j+e

1(x₀+e₁) < u^∗_j(x₀). (2.32)

Forx∈B_1/2(z_j), set ψ(x)=u^∗_j+e

1(x+e₁),χ=max(u^∗_j, ψ),ζ =min(u^∗_j, ψ). Thenζ =u^∗_j=U₁τ₋¹₁U₁= ψ=χonB_1/2(z_j)\B_r(z_j). Thus

Br(zj)

L(ζ )+L(χ ) dx=

Br(zj)

L(u^∗_j)+L(ψ)

dx (2.33)

which shows thatζ minimizes (2.23) overSr(zj)andτ1χ minimizes (2.23) overSr(zj+e1). Therefore by the definitions ofU^∗andζ,

u^∗_j(x0)ζ (x0)ψ(x0)=u^∗_j+e

1(x0+e1) (2.34)

contrary to (2.32). Thus (b) is verified. To prove that (c) is valid, suppose not. IfU^∗≡w₀, then U₁=w₀ in Rⁿ\ _j∈Zⁿ(zj)soτ_j¹U₁→v₀inL²(T (0))asj→ ∞, contrary to Proposition 2.14 forU₁. SimilarlyU^∗≡v₀. Thus (a), (b), and (c) hold andU₁satisfies (PDE). Since anyU∈M1is trivially the limit of a minimizing sequence for (2.17), the argument just given showsUsatisfies (PDE) and 2^oof the theorem is satisfied.

To prove 3^oof Theorem 2.18, observe that U1−v0 L²(T (ie₁))→0 asi→ −∞by Proposition 2.14. SinceU1

andv0are bounded inL^∞(Rⁿ), local Schauder estimates showU1−v0is bounded inC_loc^2,α(Rⁿ)for anyα∈(0,1).

Standard interpolation inequalities then yield 3^oforU1−v0and a similar argument applies toU1−w0.

To establish 4^o–5^o, we begin with 5^o. Let V , W ∈M1. If 5^o is false, setting ϕ =max(V , W ) and ψ= min(V , W ), thenϕ(z)=ψ(z)for somez∈Rⁿ. Suppose for the moment thatϕ, ψ∈Γ1. Arguing as in the proof of Proposition 2.2 [8], for alli∈Z,

T (ie₁)

L(ϕ)dx+

T (ie₁)

L(ψ)dx=

T (ie₁)

L(V )dx+

T (ie₁)

L(W )dx (2.35)

and this implies

2c₁J₁(ϕ)+J₁(ψ)=J₁(V )+J₁(W )=2c₁. (2.36)

ThusJ₁(ϕ),J₁(ψ)=c₁and by 1^o–2^o,ϕ andψare solutions of (PDE). Butϕ−ψ0,ϕ(z)=ψ(z), andϕ−ψ is a solution of the linear elliptic partial differential equation

−Φ+A(x)Φ=0 (2.37)

where

A(x)=F_u(x, ϕ(x))−F_u(x, ψ(x))

ϕ(x)−ψ(x) , ϕ(x) > ψ(x)

=F_uu x, ϕ(x)

, ϕ(x)=ψ(x).

Further writing (2.37) as

−Φ+max(A,0)Φ= −min(A,0)Φ0, (2.38)

the maximum principle impliesϕ≡ψ, a contradiction.

To verify thatϕ, ψ∈Γ1, it suffices to prove that

τ₋¹₁χχ (2.39)

forχ=ϕ, ψ. First forϕ, note that

τ₋¹₁ϕ(x)=ϕ(x₁+1, x₂, . . .)=max

V (x₁+1, x₂, . . .), W (x₁+1, x₂, . . .)

. (2.40)

Ifτ₋¹₁ϕ(x)=τ₋¹₁V (x), sinceτ₋¹₁V (x)V (x), then by (2.40), τ₋¹₁V (x)τ₋¹₁W (x)W (x).

A similar argument applies ifτ₋¹₁ϕ(x)=τ₋¹₁W (x). Hence (2.39) holds forϕ.

Next to prove (2.39) forψ, ifτ₋₁¹ ψ(x)=τ₋₁¹ V (x)andψ(x)=V (x), (2.39) is valid while ifτ₋₁¹ ψ=τ₋₁¹ V (x) andψ(x)=W (x),

τ₋¹₁ψ(x)=τ₋¹₁V (x)V (x)W (x)=ψ(x).

A similar argument obtains if the roles ofV andW are reversed. Thusϕ, ψ∈Γ₁and 5^ois proved.

To get 4^o, note that

v0U1τ₋¹₁U1w0. (2.41)

Now the maximum principle can be used exactly as in (2.37)–(2.38) to get strict inequalities in (2.41). Lastly 6^o follows from 5^oand the argument of Corollary 2.3.

The proof of Theorem 2.18 is complete. 2

3. Minimality ofU₁

As was mentioned in the introduction, Moser studied solutions of (1.1) that were minimal in the sense of Giaquinta and Guisti. In this section it will be shown thatU₁is a minimal solution of (PDE) in this sense.

Following [6],U₁is a minimal solution of (PDE) if for any bounded domainΩ⊂Rⁿwith a smooth boundary,

Ω

L(u)dx

Ω

L(U₁)dx (3.1)

for anyu∈W_loc^1,2(Rⁿ)withu=U₁inRⁿ\Ω. In other wordsU minimizes

ΩL(·)dx over the class ofW^1,2(Ω) functions havingU₁as boundary values. The proof of Theorem 2.18 shows thatU₁satisfies (3.1) whenΩ is any ball of radiusr < ¹₂. To extend this property to the more general class of boundedΩ’s with a smooth boundary requires showing thatc₀andc₁can be characterized as minimizers of functionals in broader classes of functions.

To begin, letp=(p₁, . . . , p_n)∈Nⁿand set Γ₀(p)=

u∈W_loc^1,2(Rⁿ)|uisp_i periodic inx_i, 1in , I_p(u)=

p₁

0

· · ·

pn

0

L(u)dx, c₀(p)= inf

u∈Γ₀(p)I_p(u), and

M0(p)=

u∈Γ₀(p)|Ip(u)=c₀(p) .

The proof of Proposition 2.2 shows thatM0(p)=φand is an ordered set.

Lemma 3.2.M0(p)=M0(and thereforec₀(p)=(_n

1p_i)c₀).

Proof. It suffices to show thatτ₋ⁱ₁u=u for 1in and anyu∈M0(p). If not, since M0(p)is ordered, either (i)τ₋ⁱ₁u > uor (ii)τ₋ⁱ₁u < u. If e.g. (i) occurs,

u < τ₋ⁱ₁u <· · ·< τ₋ⁱ_p

iu=u,

a contradiction. Similarly (ii) cannot occur and the lemma follows. 2

Next it will be shown that there is an analogue of Lemma 3.2 in the setting of Theorem 2.18. Let = (₂, . . . , _n)∈Nⁿ⁻¹and define

Γ₁()=

u∈W_loc^1,2(Rⁿ)|v₀uw₀anduis_i periodic inx_i, 2in . Letp∈Nⁿwithp=(p₁, )and leti∈Z. Foru∈Γ₁(), define

J_1,i^p(u)=

2−1 k₂=0

· · ·

n−1 k_n=0

T ((ip₁,k))

L(u)dx−c₀

and

J₁^p(u)=

i∈Z

J_1,i^p(u).

Observe that with slight modifications in the proof,J₁^phas the same properties onΓ₁()as doesJ₁onΓ₁given by Proposition 2.6. Set

Γ1(p)=

u∈Γ1()|uτ₋¹_p

1uandv0≡u≡w0

. ReplacingΓ1, J1, andT (0)by Γ1(p),J₁^p, and _0k

rprT (k1e1+ · · · +knen), the proof of Proposition 2.14 carries over to the current setting.

Now define c₁(p)= inf

u∈Γ1(p)J₁^p(u).

By the above observations, the argument of Theorem 2.18 (withr <¹₂min_1inp_i now permitted) applies here so M1(p)≡

u∈Γ₁(p)|J₁^p(u)=c₁(p)

is a nonempty ordered set of solutions of (PDE). The analogue of Lemma 3.2 in this setting is:

Lemma 3.3.M1(p)=M1andc₁(p)=(_n

1p_i)c₁.

Proof. It suffices to show that wheneveru∈M1(p): (i)τ₋ⁱ₁u=u, 2in, and (ii)τ₋¹₁uu. The proof of (i) is the same as that of Lemma 3.2. For (ii), observe thatτ₋¹₁u∈M1(p)which is ordered. Hence if (ii) fails,u > τ₋¹₁u so by the definition ofΓ₁(p),

uτ₋¹_p

1u < τ₋¹_p

1+1u <· · ·< u, a contradiction. 2

Remark 3.4. By (F₂), the replacement of[ip₁, ip₁+1]inT (ip₁, k)by[ip₁+j, ip₁+j+1]for anyj∈Zdoes not effect the above arguments. The same is true ifis replaced by+qfor anyq∈Rⁿ⁻¹.

Theorem 3.5. AnyU∈M1is a minimal solution of (PDE) in the sense of Giaquinta and Guisti.

Proof. To show that (3.1) is satisfied, letz∈Rⁿandr >0 such thatΩ⊂B_r(z). Set Sr(z)=

u∈W_loc^1,2(Rⁿ)|u=UinRⁿ\B_r(z) . It suffices to prove that

Br(z)

L(u)dx

Br(z)

L(U )dx (3.6)

for anyu∈Sr(z). By Lemma 3.3, M1=M1(p)for anyp∈Nⁿ. Choosepso that min_1inp_i >2r. Further exploiting Remark 3.4, it can be assumed that Ω ⊂B_r(z)⊂T (p). Hence by the proof of Theorem 2.18, U minimizes

Br(z)L(·)dxoverSr(z)and the proof is complete. 2 An immediate consequence of Theorem 3.5 is

Corollary 3.7.U1is the unique minimizer of

B_r(z)L(·)dxinSr(z).

Proof. Supposeu∈Sr(z)so that

Br(z)

L(u)dx=

Br(z)

L(U1)dx. (3.8)

Letr^∗> r. Then by (3.8),

B_r∗(z)

L(u)dx=

Br∗(z)\Br(z)

L(U1)dx+

B_r(z)

L(u)dx=

B_r∗(z)

L(U1)dx. (3.9)

Hence U₁∈Sr^∗(z) and minimizes

B_r∗(z)L(·)dx over Sr^∗(z). Again as in Lemma 2.5 of [9], the set of such

minimizers is ordered. Sinceu andU₁ belong to this set andu=U₁ inB_r∗(z)\B_r(z), u≡U₁. The proof is complete. 2

4. The general case

The goal of this section is to show how the results of Sections 1–2 together with induction and minimization arguments can be used to obtain more complex heteroclinic solutions of (PDE) corresponding to those obtained by Bangert [4] via his nonvariational approach.

To give an idea of the inductive procedure at level two, supposeM1as obtained in Theorem 1.18 satisfies a gap condition:

(∗)₁ There are adjacentv₁< w₁inM1. Define the set of functionsΓ₂via

Γ₂=

u∈W_loc^1,2(Rⁿ)|v₁uτ₋¹₁uw₁anduis 1-periodic inx₃, . . . , xn

. The renormalized functional,J₂onΓ₂is defined by

J₂(u)=

i∈Z

J_2,i(u) where

J_2,i(u)=J₁ τ₋²_iu

−c₁.

Suppose thatJ₂onΓ₂has the analogues of the properties ofJ₁onΓ₁as given by Proposition 2.6 with 3^oreplaced

by

T (₁e₁+₂e₂)

L(u)dxJ₂(u)+c₀+c₁

for any1, 2∈Z. Suppose also that Proposition 2.14 is valid with appropriate changes of sub- or superscript 1’s to 2’s. Setting

Γ₂=

u∈Γ₂|uτ₋²₁uandv₁≡u≡w₁ and

c₂= inf

u∈Γ2

J₂(u),

the analogue of Theorem 2.18 holds here providing a solution of (PDE) that is heteroclinic inx₁fromv₀tow₀and heteroclinic inx₂fromv₁tow₁. Moreover as in Theorem 3.5,U₂is a minimal solution of (PDE) in the sense of Giaquinta and Guisti.

Now setting M2=

u∈Γ₂|J₂(u)=c₂ ,

a gap condition(∗)₂can be introduced and the process continues. To carry out the induction argument properly, let m < nand assume the gap condition:

(∗)i There are adjacentvi< wi inMi holds fori=0, . . . , m−1. Let Γ_i=

u∈W_loc^1,2(Rⁿ)|v_i−1uτ₋^j₁uw_i−1, 1j < ianduis 1-periodic inx_i+1, . . . , x_n (4.1)_i for 1im. Theith renormalized functional,J_i(u), is given by

J_i(u)=

p∈Z

J_i,p(u) (4.2)_i

where

Ji,p(u)=Ji−1

τ₋ⁱ_pu

−ci−1. (4.3)i

Suppose thatJ_i onΓ_i possesses the following properties for 1im:

Proposition 4.4_i. Foru∈Γ_i, 1^o J_i,p(u)0 for allp∈Z. 2^o J_i(u)0.

3^o

T (i

1qeq)L(u)dxJ_i(u)+_i−1

0 c_q. 4^o J_i is weakly lsc (inW_loc^1,2(Rⁿ)) onΓ_i.

For 1im, set Γi=

u∈Γi|uτ₁ⁱuandvi−1≡u≡wi−1

(4.5)i

and assume (with the understanding that₀

1_qe_q=0):

Proposition 4.6i. Ifu∈Γi andJi(u) <∞, then asj → ∞,τ_jⁱu→vi−1weakly inW^1,2(T (i−1

1 qeq))for all ₁, . . . , i−1∈Zand asj→ −∞,τ_jⁱu→wi−1weakly inW^1,2(T (i−1

1 qeq))for all₁, . . . , i−1∈Z. Finally define

c_i= inf

u∈Γi

J_i(u), 1im, (4.7)_i

and assume:

Theorem 4.8_i. LetF satisfy (F₁)–(F₃) and let(∗)_i hold. Then 1^o There is aUi∈Γi such thatJi(Ui)=ci.

2^o Any suchUi is a classical solution of (PDE).