1 Introduction
In this paper, we consider the following prescribed mean
curvature problem on the torus πn:=βn/β€n:
(1.1)-divβ‘(ββ‘u1+|ββ‘u|2)=Ξ½β’(ββ‘u)β
gβ’(x,uβ’(x))βonΒ β’πn,
where Ξ½ is the unit normal vector of u, that is,
Ξ½β’(z)=11+|z|2β’(-z,1).
The vector field gβ’(x,xn+1):πnΓβββn+1 is given, and we
seek a solution u satisfying (1.1).
The left-hand side of (1.1) represents the mean curvature of the graph of u,
and the right-hand side is the normal component of the vector field g on the graph.
In the case of Dirichlet conditions of a bounded domain Ξ©ββn,
prescribed mean curvature problems have been studied by numerous researchers.
Bergner [3] solved the Dirichlet problem in the case where the right-hand side of (1.1)
is H=Hβ’(x,u,Ξ½β’(ββ‘u)) under the assumptions of boundedness (|H|<β),
monotonicity (βn+1β‘Hβ₯0), and convexity of Ξ©.
Under the same conditions for the function H,
Marquardt [9] imposed a condition on ββ‘Ξ© depending on H
that guarantees the existence of solutions even for a domain Ξ© that is not necessarily convex.
In [13], we proved the existence of a solution only under the condition that
the Sobolev norm of H is sufficiently small.
In the case of a compact Riemannian manifold,
Aubin [2] solved the linear elliptic problem -βiβ‘[aiβ’jβ’(x)β’βjβ‘u]=Hβ’(x)
if the integrated value of H is zero. The assumption of the
integrated value plays an important role in the existence of solutions to elliptic equations
on a compact Riemannian manifold.
Denny [4] solved the quasilinear elliptic problem -divβ‘(aβ’(uβ’(x))β’ββ‘u)=Hβ’(x) on the torus πn with n=2,3.
Prescribed mean curvature problems on the one-dimensional torus
(uβ²1+(uβ²)2)β²=Hβ’(x,u,uβ²)
have been investigated
for a wide variety of conditions H (we refer to, for example, [5, 7, 8, 10, 11, 14]).
As we noted in [13], the motivation for the present study comes from a singular perturbation problem, and we proved the following in [12]. Suppose a constant Ξ΅>0 and functions ΟΞ΅βW1,2 and gΞ΅βW1,p, with p>n+12, satisfy
-Ξ΅β’Ξβ’ΟΞ΅+Wβ²β’(ΟΞ΅)Ξ΅=Ξ΅β’ββ‘ΟΞ΅β
gΞ΅,
β«(Ξ΅β’|ββ‘ΟΞ΅|22+Wβ’(ΟΞ΅)Ξ΅)β’πx+β₯gΞ΅β₯W1,pβ’(Ξ©~)β€C,
where W is a double-well potential such as Wβ’(Ο)=(1-Ο2)2.
Then the interface {ΟΞ΅=0} converges locally in the Hausdorff distance to a
surface having a mean curvature given by Ξ½β
g as Ξ΅β0. Here, Ξ½ is the unit normal vector of the surface, and g is the weak W1,p limit of gΞ΅.
If the surface is represented locally as a graph of a function u on πn,
we can observe that u satisfies (1.1).
In this paper, we prove the existence of solutions to (1.1) assuming
that the Sobolev norm of g is sufficiently small, gn+1 for the (n+1)-st component is monotonous,
and the integrated value of gn+1 is zero.
The following theorem is the main result.
Theorem 1.1.
Fix n+12<p<n+1 and q=nβ’pn+1-p.
Then there exists a constant Ξ΅1=Ξ΅1β’(n,p)>0
with the following property:
If Ξ΅<Ξ΅1, and
g=(g1,β¦,gn,gn+1)=(gβ²,gn+1)βW1,pβ’(πnΓ(-1,1);βn+1)
satisfies the relations
(1.2)β₯gβ₯W1,pβ’(πnΓ(-1,1))<Ξ΅23,
(1.3)βn+1β‘gn+1β’(x,xn+1)>Ξ΅+Ξ΅12β’|βn+1β‘gβ²β’(x,xn+1)|,
(1.4)β«πngn+1β’(x,0)=0,
then there exists a function uβW2,qβ’(Tn) such that
(1.5)-divβ‘(ββ‘u1+|ββ‘u|2)=Ξ½β’(ββ‘u)β
gβ’(x,uβ’(x))βonΒ β’πn.
Moreover, the following inequality holds:
β₯u-β«πnuβ’(y)β’πyβ₯W2,qβ’(πn)β€Ξ΅12.
Assumptions (1.2) and (1.3) guarantee the existence and uniqueness
of solutions to the linearized problem of (1.1) where a given function depends on ββ‘u.
Equation (1.4) is necessary for the existence of solutions to elliptic equations on the torus.
To the best of our knowledge, prescribed mean curvature problems on the torus in the general dimension have been insufficiently studied. However, we have proved the existence of the solution under natural assumptions.
The following is the method of proof. First, we find the conditions of H for the linearized problem of (1.1), i.e.
-divβ‘(ββ‘u1+|ββ‘v|2)=H,
to have a unique solution. If we add a suitable constant term for any v,
the function Ξ½β’(ββ‘v)β
gβ’(x,vβ’(x))
satisfies the conditions.
By estimating the norm of this solution with g, the mapping Tβ’(v)=u has a fixed point
using a fixed-point theorem, and Theorem 1.1 follows.
2 Proof of Theorem 1.1
A theorem that holds in the Euclidean space also holds on a torus, as we consider a function on a torus to be a periodic function in the Euclidean space.
Let Xβ’(πn) be a function space on πn.
We define a subspace Xaveβ’(πn)βXβ’(πn) as
Xave:={wβX:β«πnw=0}.
Theorem 2.1.
Suppose vβC1β’(Tn) and HβLave2β’(Tn).
Then there exists a unique function uβWave1,2β’(Tn) such that
β«πnββ‘uβ
ββ‘Ο1+|ββ‘v|2=β«πnHβ’Ο
for all ΟβW1,2β’(Tn).
Proof.
We define a function B:Wave1,2β’(πn)ΓWave1,2β’(πn)ββ by
Bβ’[w1,w2,v]:=β«πnββ‘w1β
ββ‘w21+|ββ‘v|2.
By the HΓΆlder inequality, we obtain
|Bβ’[w1,w2,v]|β€β«πn|ββ‘w1|β’|ββ‘w2|
β€β₯ββ‘w1β₯L2β’(πn)β’β₯ββ‘w2β₯L2β’(πn)
(2.1)β€β₯w1β₯W1,2β’(πn)β’β₯w2β₯W1,2β’(πn).
Using the PoincarΓ© inequality, we have
|Bβ’[w,w,v]|β₯11+β₯vβ₯C1β’(πn)2β’β₯ββ‘wβ₯L2β’(πn)2
(2.2)β₯11+β₯vβ₯C1β’(πn)2β’β₯ββ‘wβ₯W1,2β’(πn)2.
By (2.1), (2.2), and the LaxβMilgram theorem,
for any HβLave2β’(πn),
there exists a unique function
uβWave1,2β’(πn)
such that
(2.3)β«πnββ‘uβ
ββ‘Ο1+|ββ‘v|2=β«πnHβ’Ο
for all ΟβWave1,2β’(πn).
For any ΟβW1,2β’(πn),
we define cΟ:=β«πnΟ and Ο~:=Ο-cΟβWave1,2β’(πn).
By (2.3) and HβLave2β’(πn), we obtain
β«πnββ‘uβ
ββ‘Ο1+|ββ‘v|2=β«πnββ‘uβ
ββ‘Ο~1+|ββ‘v|2=β«πnHβ’Ο~=β«πnHβ’Ο.
Thus, Theorem 2.1 follows.
β
We define a mollifier as follows:
Ξ·β’(x):={Cβ’expβ‘(1|x|2-1)forΒ β’|x|<1,0forΒ β’|x|β₯1,
where the constant C>0 is selected such that β«βn+1Ξ·=1. We define
Ξ·Ξ»β’(x):=1Ξ»nβ’Ξ·β’(xΞ»).
For any fβL2β’(πnΓ(-1,1)) and xn+1β(-1+Ξ»,1-Ξ»),
fΞ»β’(x,xn+1):=β«πnΓ(-1,1)Ξ·Ξ»β’(x-y,xn+1-yn+1)β’fβ’(y,yn+1)β’πy
=β«Bn+1β’(0,Ξ»)Ξ·Ξ»β’(y,yn+1)β’fβ’(x-y,xn+1-yn+1)β’πy,
where Bn+1β’(x,Ξ») is an open ball with center x and radius Ξ» in πnΓβ.
Moreover, for any
gβW1,pβ’(πnΓ(-1,1);βn+1),
we define gΞ»:=(gΞ»1,β¦,gΞ»n,gΞ»n+1)=(gΞ»β²,gΞ»n+1).
Lemma 2.2.
Fix Ξ²1>0 and 0<Ξ»<1. Suppose vβC1β’(Tn) satisfies β₯vβ₯C1β’(Tn)<Ξ²1, and
gβW1,pβ’(πnΓ(-1,1);βn+1)
satisfies
βn+1β‘gn+1β’(x,xn+1)>Ξ²1β’|βn+1β‘gβ²β’(x,xn+1)|.
For any positive constant c0>0, if vβ’(Tn)+c0β(-1+Ξ»,1-Ξ»), then
β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v)<β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v+c0).
Proof.
From the assumptions, we compute
β«πnΞ½β’(ββ‘v)β
(gΞ»β’(x,v+c0)-gΞ»β’(x,v))
=β«πn11+|ββ‘v|2β’β«vv+c0-ββ‘vβ
βn+1β‘gΞ»β²β’(x,t)+βn+1β‘gΞ»n+1β’(x,t)β’dβ’t
β₯β«πn11+|ββ‘v|2β’β«vv+c0-Ξ²1β’|βn+1β‘gΞ»β²β’(x,t)|+βn+1β‘gΞ»n+1β’(x,t)β’dβ’t
β₯β«πn11+|ββ‘v|2β’β«vv+c0β«πnΓ(-1,1)Ξ·Ξ»β’(x-y,t-yn+1)β’{-Ξ²1β’|βn+1β‘gβ²β’(y,yn+1)|+βn+1β‘gn+1β’(y,yn+1)}β’πt
>0.
Lemma 2.2 follows.
β
Lemma 2.3.
Suppose gβW1,p(TnΓ(-1,1)
and vβC1β’(Tn) with β₯vβ₯C1β’(Tn)β€716.
Let q=nβ’pn+1-p. Then there exists a constant c1=c1β’(n,p)>0 such that,
if Ξ»<18,
β₯gΞ»β’(β
,vβ’(β
))β₯Lqβ’(πn)β€c1β’β₯gβ₯W1,pβ’(πnΓ(-1,1)).
Proof.
By the same proof as in [13, Lemma 2.3], we obtain
(2.4)β₯gΞ»β’(β
,vβ’(β
))β₯Lqβ’(πn)β€c2β’β₯gΞ»β₯W1,pβ’(πnΓ(-78,78)),
where c2=c2β’(n,p)>0.
Using the HΓΆlder inequality, we obtain
β«πnΓ(-78,78)|gΞ»|pβ’πxβ€β«πnΓ(-78,78)(β«Bn+1β’(x,Ξ»)Ξ·Ξ»1-1p+1pβ’(x-y,xn+1-yn+1)β’|gβ’(y,yn+1)|β’πy)pβ’πx
β€β«πnΓ(-78,78)(β«Bn+1β’(x,Ξ»)Ξ·Ξ»β’(x-y,xn+1-yn+1)β’|gβ’(y,yn+1)|pβ’πy)β’πx
β€β«πnΓ(-1,1)|gβ’(y,yn+1)|pβ’(β«Bn+1β’(y,Ξ»)Ξ·Ξ»β’(x-y,xn+1-yn+1)β’πx)β’πy
(2.5)=β«πnΓ(-1,1)|gβ’(y,yn+1)|pβ’πy.
We can show that
β₯ββ‘gΞ»β₯Lpβ’(πnΓ(-78,78))β€β₯ββ‘gβ₯Lpβ’(πnΓ(-1,1))
in the exact same manner, and
Lemma 2.3 follows by (2.4) and (2.5).
β
Theorem 2.4.
Suppose vβC1β’(Tn) and
gβW1,pβ’(πnΓ(-1,1);βn+1).
Then there exist constants Ξ΅2=Ξ΅2β’(n,p)>0 such that, if Ξ»<18, Ξ΅<Ξ΅2, and β₯vβ₯C1β’(Tn)β€Ξ΅1/2, then
g satisfies (1.2)β(1.4). Then
there exist a unique function uβWave1,2β’(Tn) and
a unique constant -14<cv<14 such that
(2.6)β«πnββ‘uβ
ββ‘Ο1+|ββ‘v|2=β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v+cv)β’Ο
for all ΟβW1,2β’(Tn).
Proof.
We define
Fβ’(t):=β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v+t).
The function F is continuous. Suppose that Ξ΅<1162. We will consider that
the domain of F is [-14,14].
By the mean value theorem,
there exists a constant c3=c3β’(n,p)>0 such that
Fβ’(14)=β«πn(Ξ½β’(ββ‘v)-Ξ½β’(0)+Ξ½β’(0))β
gΞ»β’(x,v+14)
(2.7)β₯-c3β’β₯vβ₯C1β’(πn)β’β₯gΞ»β’(β
,vβ’(β
)+14)β₯Lqβ’(πn)+β«πngΞ»n+1β’(x,v+14).
By Lemma 2.3 and β₯v+14β₯C1β’(πn)β€516, we obtain
β₯gΞ»β’(β
,vβ’(β
)+14)β₯Lqβ’(πn)β€c1β’β₯gβ₯W1,pβ’(πnΓ(-1,1)).
By (1.3) and (1.4),
there exists a constant c4=c4β’(n)>0 such that
β«πngΞ»n+1β’(x,v+14)=β«πnβ«Bn+1β’(0,Ξ»)Ξ·Ξ»β’(y,yn+1)β’gn+1β’(x-y,v+14-yn+1)β’πyβ’πx
>β«πnβ«Bn+1β’(0,Ξ»)Ξ·Ξ»β’(y,yn+1)β’gn+1β’(x-y,116)β’πyβ’πx
>β«πnβ«Bn+1β’(0,Ξ»)Ξ·Ξ»β’(y,yn+1)β’(gn+1β’(x-y,0)+Ξ΅16)β’πyβ’πx
(2.8)>c416β’Ξ΅.
By (1.2), (2.7)β(2.8), and β₯vβ₯C1β’(πn)<Ξ΅1/2, if
Ξ΅<(c416β’c1β’c3)6=:Ξ΅2(n,p),
then
Fβ’(14)>-c1β’c3β’β₯vβ₯C1β’(πn)β’β₯gΞ»β₯W1,pβ’(πnΓ(-1,1))+c416β’Ξ΅
>-c1β’c3β’Ξ΅76+c416β’Ξ΅
>Ξ΅β’(-c1β’c3β’Ξ΅16+c416)
>0.
Similarly, we can show that
Fβ’(-14)<0.
By Lemma 2.2 and the mean value theorem,
there exists a unique constant -14<cv<14 that satisfies Fβ’(cv)=0.
By using Theorem 2.1, Theorem 2.4 follows.
β
Let us define an operator T:πβ’(s)βWave1,2β’(πn)Γ[-14,14] by Tβ’(v)=(T1β’(v),T2β’(v)):=(u,cv) that satisfies
(2.6), where
πβ’(s):={wβWave2,qβ’(πn):β₯wβ₯W2,qβ’(πn)β€s}.
Theorem 2.5.
There exist constants Ξ΅3=Ξ΅3β’(n,p)>0 and
c5=c5β’(n,p)>0 such that,
if Ξ»<18, Ξ΅<minβ‘{Ξ΅2,Ξ΅3}, vβAβ’(Ξ΅1/2),
and gβW1,pβ’(TnΓ(-1,1);Rn+1) satisfies (1.2)β(1.4),
then
β₯T1β’(v)β₯W2,qβ’(πn)β€c5β’β₯gβ₯W1,pβ’(πnΓ(-1,1)).
Proof.
We first assume that vβCββ’(πn)β©πβ’(Ξ΅1/2).
Using [6, Corollary 8.11], we obtain T1β’(v)βCββ’(πn). Thus, we can
rewrite (2.6) as
Ξβ’T1β’(v)1+|ββ‘v|2+ββ‘T1β’(v)β
ββ‘(11+|ββ‘v|2)=-Ξ½β’(ββ‘v)β
gΞ»β’(x,v+T2β’(v)).
Using [6, Theorem 9.11],
we find that there exists a constant
c6=c6β’(n,p)>0 such that
β₯T1(v)β₯W2,qβ’(πn)β€c6(β₯T1(v)β₯Lqβ’(πn)+β₯Ξ½(βv)β
gΞ»(x,v+T2(v))β₯Lqβ’(πn)
(2.9)+β₯βT1(v)β
β(11+|ββ‘v|2)β₯Lqβ’(πn)).
Using Lemma 2.3, we obtain
(2.10)β₯Ξ½β’(ββ‘v)β
gΞ»β’(x,v+T2β’(v))β₯Lqβ’(πn)β€c1β’β₯gβ₯W1,pβ’(πnΓ(-1,1)).
Using the Sobolev inequality,
we find that there exists a constant
c7=c7β’(n,p)>0 such that
β₯ββ‘T1β’(v)β
ββ‘(11+|ββ‘v|2)β₯Lqβ’(πn)β€β₯T1β’(v)β₯C1β’(πn)β’β₯ββ‘(11+|ββ‘v|2)β₯Lqβ’(πn)
(2.11)β€c7β’β₯T1β’(v)β₯W2,qβ’(πn)β’β₯vβ₯W2,qβ’(πn).
Next, we estimate the term β₯T1β’(v)β₯Lqβ’(πn). If qβ€2, then, by (2.2) and
Lemma 2.3, we obtain
β₯T1β’(v)β₯Lqβ’(πn)β€c8β’(n,p)β’β₯T1β’(v)β₯L2β’(πn)
β€c9β’(n,p)β’Bβ’[T1β’(v),T1β’(v),v]12.
=c9β’(β«πnββ‘T1β’(v)β
ββ‘T1β’(v)1+|ββ‘v|2)12
=c9β’(β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v+T2β’(v))β’T1β’(v))12
β€c10β’(n,p)β’β₯gβ₯W1,pβ’(πn)12β’β₯T1β’(v)β₯Lββ’(πn)12
(2.12)β€c11β’(n,p)β’β₯gβ₯W1,pβ’(πn)+14β’c6β’β₯T1β’(v)β₯W2,qβ’(πn).
If q>2, by (2.12) and the RieszβThorin theorem, we obtain
β₯T1β’(v)β₯Lqβ’(πn)β€β₯T1β’(v)β₯L2β’(πn)1qβ’β₯T1β’(v)β₯L2β’(πn)1-1q
β€c12β’(n,p)β’β₯gβ₯W1,pβ’(πn)12β’qβ’β₯T1β’(v)β₯Lββ’(πn)12β’q+1-1q
(2.13)β€c13β’(n,p)β’β₯gβ₯W1,pβ’(πn)+14β’c6β’β₯T1β’(v)β₯W2,qβ’(πn).
By (2.9)β(2.13), there exists a constant c14=c14β’(n,p)>0
such that
β₯T1β’(v)β₯W2,qβ’(πn)β€c14β’(β₯gβ₯W1,pβ’(πnΓ(-1,1))+β₯T1β’(v)β₯W2,qβ’(πn)β’β₯vβ₯W2,qβ’(πn))+14β’β₯T1β’(v)β₯W2,qβ’(πn).
If Ξ΅<116β’c142, we obtain
(2.14)β₯T1β’(v)β₯W2,qβ’(πn)β€2β’c14β’β₯gβ₯W1,pβ’(πnΓ(-1,1)).
For the general case of vβW2,qβ’(πn),
suppose that {vm}mβββCββ’(πn) converges to v in the sense of C1β’(πn).
By (2.14), there exists a subsequence
{vmk}kβββ{vm}mββ
such that T1β’(vmk) converges to a function wββW2,qβ’(πn)
in the sense of C1β’(πn), and
T2β’(vmk) converges to a constant dββ[-14,14].
For any ΟβW1,2β’(πn), we obtain
β«πnΞ½β’(ββ‘v)β
gΞ»β’(x,v+dβ)β’Ο-Ξ½β’(ββ‘vmk)β
gΞ»β’(x,vmk+T2β’(vmk))β’Ο
β€β«πn|Ο|β’|Ξ½β’(ββ‘v)-Ξ½β’(ββ‘vmk)|β’|gΞ»β’(x,vmk+T2β’(vmk))|+β«πn|Ο|β’|β«vmk+T2β’(vmk)v+dββn+1β‘gΞ»β’(x,s)|
(2.15)β0β(kββ)
and
β«πnββ‘wββ
ββ‘Ο1+|ββ‘v|2-ββ‘T1β’(vmk)β
ββ‘Ο1+|ββ‘vmk|2
β€β«πn(ββ‘wβ-ββ‘T1β’(vmk))β
ββ‘Ο1+|ββ‘v|2+β«πn(ββ‘T1β’(vmk)β
ββ‘Ο)β’(11+|ββ‘v|2-11+|ββ‘vmk|2)
(2.16)β0β(kββ).
By (2.15) and (2.16), we obtain
β«πnββ‘wββ
ββ‘Ο1+|ββ‘v|2-Ξ½β’(ββ‘v)β
gΞ»β’(x,v+dβ)β’Ο
=limkβββ‘β«πnββ‘T1β’(vmk)β
ββ‘Ο1+|ββ‘vmk|2-Ξ½β’(ββ‘vmk)β
gΞ»β’(x,vmk+T2β’(vmk))β’Ο
(2.17)=0,
that is, Tβ’(v)=(wβ,dβ). By (2.14) and (2.17), Theorem 2.5 follows.
β
Next, we provide the fixed-point theorem, which is needed later ([1, Theorem 1]).
An operator T:XβA is considered weakly sequentially continuous if,
for every sequence {xm}mβββX and xββX such that
xm weakly converges to xβ, Tβ’(xm) weakly converges to Tβ’(xβ).
Theorem 2.6.
Let X be a metrizable, locally convex topological vector space and let Ξ© be a weakly compact convex subset of X. Then any weakly
sequentially continuous map T:Ξ©βΞ© has a fixed point.
We first prove Theorem 1.1 in the case of gΞ».
Theorem 2.7.
There exists a constant Ξ΅4=Ξ΅4β’(n,p)>0 such that,
if Ξ»<18 and Ξ΅<Ξ΅4, then
gβW1,pβ’(πnΓ(-1,1);βn+1)
satisfies (1.2)β(1.4).
Then there exists a function uΞ»βW2,qβ’(Tn) such that
(2.18)-divβ‘(ββ‘uΞ»1+|ββ‘uΞ»|2)=Ξ½β’(ββ‘uΞ»)β
gΞ»β’(x,uΞ»β’(x))βonΒ β’πn.
Proof.
The set W2,qβ’(πn) is a metrizable, locally convex topological vector space, and
the set πβ’(Ξ΅1/2) is a weakly compact convex subset of W2,qβ’(πn).
By (1.2) and Theorem 2.5, if
Ξ΅<min{Ξ΅2,Ξ΅3,c5-6}=:Ξ΅4,
we have
β₯T1β’(v)β₯W2,qβ’(πn)β€c5β’β₯gβ₯W1,pβ’(πnΓ(-1,1))
β€c5β’Ξ΅16β’Ξ΅12
(2.19)β€Ξ΅12βfor anyΒ β’vβπβ’(Ξ΅12),
that is,
T1β’(πβ’(Ξ΅1/2))βπβ’(Ξ΅1/2).
Suppose that {vm}mββ weakly converges to vβ in the sense of W2,qβ’(πn).
According to Theorem 2.5, there exists a subsequence {vmk}kβββ{vm}mββ
such that T1β’(vmk) weakly converges to a function wββW2,qβ’(πn)
in the sense of W2,qβ’(πn), and
T2β’(vmk) converges to a constant dββ[-14,14].
By the same argument (2.15)β(2.17), for any ΟβW2,qβ’(πn),
β«πnββ‘wββ
ββ‘Ο1+|ββ‘vβ|2-Ξ½β’(ββ‘vβ)β
gΞ»β’(x,vβ+dβ)β’Ο=0,
that is, we obtain limkβββ‘T1β’(vmk)=T1β’(vβ)
by the uniqueness of solution of Theorem 2.4.
Therefore, every convergent subsequence of {T1β’(vm)}
converges to T1β’(vβ), and T1 is a weakly sequentially continuous map.
Using Theorem 2.6, we obtain a function
vΞ»βWave2,qβ’(πn) satisfying
-divβ‘(ββ‘vΞ»1+|ββ‘vΞ»|2)=Ξ½β’(ββ‘vΞ»)β
gΞ»β’(x,vΞ»β’(x)+T2β’(vΞ»))βonΒ β’πn,
that is, uΞ»:=vΞ»+T2β’(vΞ»)βW2,qβ’(πn) satisfying (2.18).
β
Proof of Theorem 1.1.
Suppose uΞ»βW2,qβ’(πn) satisfies (2.18).
By Theorem 2.5, there exists a convergent subsequence
{uΞ»k}kβββ{uΞ»}0<Ξ»<18
with a limit uββW2,qβ’(πn) in the sense of C1β’(πn) and Ξ»kβ0. We show that uβ satisfies (1.5). For any ΟβW1,2β’(πn), we obtain
β«πn-divβ‘(ββ‘uΞ»k1+|ββ‘uΞ»k|2-ββ‘uβ1+|ββ‘uβ|2)β’Ο=β«πn(ββ‘uΞ»k1+|ββ‘uΞ»k|2-ββ‘uβ1+|ββ‘uβ|2)β
ββ‘Ο
(2.20)β0.
Using Lemma 2.3, we have
β«πnΞ½β’(ββ‘uΞ»k)β
gΞ»kβ’(x,uΞ»k)-Ξ½β’(ββ‘uβ)β
gβ’(x,uβ)
=β«πn(Ξ½β’(ββ‘uΞ»k)-Ξ½β’(ββ‘uβ))β
gΞ»kβ’(x,uΞ»k)+β«πnΞ½β’(ββ‘uβ)β
(gΞ»kβ’(x,uΞ»k)-gβ’(x,uΞ»k))
βββ+β«πnΞ½β’(ββ‘uβ)β
(gβ’(x,uΞ»k)-7β’gβ’(x,uβ))
=c1β’β₯Ξ½β’(ββ‘uΞ»k)-Ξ½β’(ββ‘uβ)β₯C0β’(πn)β’β₯gΞ»kβ₯W1,pβ’(πnΓ(-1,1))+c1β’β₯gΞ»k-gββ₯W1,pβ’(πnΓ(-1,1))+β«πn|β«uβuΞ»kβn+1β‘gβ’(x,s)|
(2.21)β0.
By (2.20) and (2.21), we obtain
β«πn-divβ‘(ββ‘uβ1+|ββ‘uβ|2)β’Ο-Ξ½β’(ββ‘uβ)β
gβ’(x,uβ)β’Ο
=limkβββ‘β«πn-divβ‘(ββ‘uΞ»k1+|ββ‘uΞ»k|2)β’Ο-Ξ½β’(ββ‘uΞ»k)β
gΞ»kβ’(x,uΞ»k)β’Ο
β0.
Thus, uβ satisfies (1.5)
using the fundamental lemma of the calculus of variations.
By (2.19), we obtain
β₯uβ-β«πnuββ’(y)β’πyβ₯W2,qβ’(πn)β€Ξ΅12,
and Theorem 1.1 follows.
β
