Boundary Value Problems
Volume 2007, Article ID 87104,9pages doi:10.1155/2007/87104
Research Article
Generalizations of the Lax-Milgram Theorem
Dimosthenis Drivaliaris and Nikos YannakakisReceived 12 December 2006; Revised 8 March 2007; Accepted 19 April 2007 Recommended by Patrick J. Rabier
We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In partic- ular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations.
Copyright © 2007 D. Drivaliaris and N. Yannakakis. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited.
1. Introduction
The following generalization of the Lax-Milgram theorem was proved recently by An et al.
in [1].
Theorem 1.1. LetX be a reflexive Banach space overR, let{Xn}n∈Nbe an increasing se- quence of closed subspaces ofXandV=
n∈NXn. Suppose that
A:X×V−→R (1.1)
is a real-valued function onX×Vfor which the following hold:
(a)An=A|Xn×Xnis a bounded bilinear form, for alln∈N; (b)A(·,v) is a bounded linear functional onX, for allv∈V;
(c)Ais coercive onV, that is, there existsc >0 such that
A(v,v)≥cv2, (1.2)
for allv∈V.
Then, for each bounded linear functionalv∗onV, there existsx∈Xsuch that A(x,v)=
v∗,v, (1.3)
for allv∈V.
In this paper our aim is to prove a linear extension and a nonlinear extension of Theorem 1.1. In the linear case, we use a variant of a theorem due to Hayden [2,3], and thus manage to substitute the coercivity condition in (c) of the previous theorem with a more general inf-sup condition. In the nonlinear case, we appropriately modify the notion of typeM operator and use a surjectivity result for monotone, hemicontinu- ous, coercive operators. We also present two examples to illustrate the applicability of our results.
All Banach spaces considered are overR. Given a Banach spaceX,X∗will denote its dual and·,·will denote their duality product. Moreover, ifM is a subset ofX, then M⊥will denote its annihilator inX∗and ifNis a subset ofX∗, then⊥Nwill denote its preannihilator inX.
2. The linear case
To prove our main result for the linear case, we need the following lemma which is a variant of [2, Theorem 12] and [3, Theorem 1].
Lemma 2.1. LetXbe a reflexive Banach space, letY be a Banach space and let
A:X×Y−→R (2.1)
be a bounded, bilinear form satisfying the following two conditions:
(a)Ais nondegenerate with respect to the second variable, that is, for eachy∈Y\ {0}, there existsx∈XwithA(x,y)=0;
(b) there existsc >0 such that sup
y=1
A(x,y)≥cx, (2.2)
for allx∈X.
Then, for everyy∗∈Y∗, there exists a uniquex∈Xwith A(x,y)=
y∗,y, (2.3)
for ally∈Y.
Proof. LetT:X→Y∗withTx,y =A(x,y), for allx∈X and ally∈Y. Obviously ,T is a bounded linear map. Since, by (b),Tx ≥cx, for allx∈X,T is one to one. To complete the proof, we need to show thatTis onto.
SinceAis nondegenerate with respect to the second variable, we have that
⊥T(X)=
y∈Y|A(x,y)=0,∀x∈X= {0}. (2.4)
Hence
⊥
T(X) ⊥=Y∗, (2.5)
and so by [4, Proposition 2.6.6],
T(X)w∗=Y∗. (2.6)
Thus to show thatTmapsXontoY∗, we need to prove thatT(X) isw∗-closed inY∗. To see that, let{Txλ}λ∈Λbe a net inT(X) and lety∗be an element ofY∗such that
Txλ−→w∗ y∗. (2.7)
Without loss of generality, we may assume, using the special case of the Krein-ˇSmulian theorem onw∗-closed linear subspaces (see [4, Corollary 2.7.12]), the proof of which is originally due to Banach [5, Theorem 5, page 124] for the separable case and due to Dieudonn´e [6, Theorem 23] for the general case, that{Txλ}λ∈Λis bounded. Thus, since Tx ≥cxfor allx∈X, the net{xλ}λ∈Λis also bounded. Hence, sinceXis reflexive, there exist a subnet{xλμ}μ∈Mand an elementxofXsuch that{xλμ}μ∈Mconverges weakly tox. Since T isw−w∗continuous, Txλμw→∗Tx. HenceTx=y∗, and soT(X) isw∗-
closed.
Remark 2.2. An alternative proof of the previous lemma can be obtained using the closed range theorem.
We are now in a position to prove our main result for the linear case.
Theorem 2.3. LetXbe a reflexive Banach space, letYbe a Banach space, letΛbe a directed set, let{Xλ}λ∈Λ be a family of closed subspaces ofX, let{Yλ}λ∈Λ be an upwards directed family of closed subspaces ofY, and letV=
λ∈ΛYλ. Suppose that
A:X×V−→R (2.8)
is a function for which the following hold:
(a)Aλ=A|Xλ×Yλis a bounded bilinear form, for allλ∈Λ;
(b)A(·,v) is a bounded linear functional onX, for allv∈V;
(c)Aλis nondegenerate with respect to the second variable, for allλ∈Λ; (d) there existsc >0 such that for allλ∈Λ,
sup
y∈Yλ,y=1
Aλ(x,y)≥cx, (2.9)
for allx∈Xλ.
Then, for each bounded linear functionalv∗onV, there existsx∈Xsuch that A(x,v)=
v∗,v, (2.10)
for allv∈V.
Proof. Letv∗∈V∗, and for eachλ∈Λ, letvλ∗=v∗|Yλ. For allλ∈Λ,v∗λ is a bounded linear functional onYλ. By hypothesis, for allλ∈Λ,Aλ is a bounded bilinear form on Xλ×Yλsatisfying the two conditions ofLemma 2.1. Since for allλ∈Λ,Xλis a reflexive Banach space, we get that for eachλ∈Λ, there exists a uniquexλsuch thatAλ(xλ,y)= vλ∗,y, for ally∈Yλ. SinceAsatisfies condition (d), we get that for allλ∈Λ,
cxλ ≤ sup
y∈Yλ,y=1
Aλ(xλ,y)= sup
y∈Yλ,y=1
v∗λ,y≤ v∗. (2.11)
So{xλ}λ∈Λ is a bounded net inX. SinceX is reflexive, there exist a subnet{xλμ}μ∈M of {xλ}λ∈ΛandxinXsuch that{xλμ}μ∈Mconverges weakly tox.
We are going to prove that A(x,v)= v∗,v, for allv∈V. Takev∈V. Then there exists someλ0∈Λwithv∈Yλ0. Since{xλμ}μ∈Mis a subnet of{xλ}λ∈Λ, there exists some μ0∈Mwithλμ0≥λ0. Hence, since the family{Yλ}λ∈Λis upwards directed,
v∈Yλμ, (2.12)
for allμ≥μ0. Thus, for allμ≥μ0,
Aλμxλμ,v =
v∗λμ,v. (2.13)
Therefore
μlim∈MAxλμ,v =
v∗,v. (2.14)
SinceA(·,v) is a bounded linear functional onX,
μlim∈MAxλμ,v =A(x,v). (2.15)
HenceA(x,v)= v∗,v.
The following example illustrates the possible applicability ofTheorem 2.3.
Example 2.4. Leta∈C1(0, 1) be a decreasing function with limt→0a(t)= ∞anda(t)≥0, for allt∈(0, 1). We will establish the existence of a solution for the following Cauchy problem:
u+a(t)u=f a.e. on (0, 1),
u(0)=0, (2.16)
where f ∈L2(0, 1).
LetX= {u∈H1(0, 1)|u(0)=0}be equipped with the normu =(01|u|2dt)1/2, which is equivalent to the original Sobolev norm, andY=L2(0, 1). Note thatXis a re- flexive Banach space, being a closed subspace ofH1(0, 1). Let{αn}n∈N be a decreasing sequence in (0, 1) with limn→∞αn=0. Define
Xn=
u∈H1αn, 1 |uαn =0, Yn=L2αn, 1 (2.17)
(we can considerXnand Yn as closed subspaces ofX and Y, resp., by extending their elements by zero outside (αn, 1)). Also letV=∞
n=1Yn. LetA:X×V→Rbe the bilinear map defined by
A(u,v)= 1
0uv dt+ 1
0a(t)uv dt. (2.18)
Ais well defined andA(·,v) is a bounded linear functional onXfor anyv∈V. LetAn=A|Xn×Yn.Anbe a bounded bilinear form since
An(u,v)≤
1 +Mn uXnvYn, (2.19) whereMnis the bound ofaon [αn, 1]. It should be noted thatAis not bounded on the whole ofX×V.
To show thatAnis nondegenerate, letv∈Ynand assume thatAn(u,v)=0 for allu∈ Xn, that is,
1 αn
u+a(t)u v dt=0, ∀u∈Xn. (2.20)
It is easy to see that the above implies that 1
αn
wv dt=0, (2.21)
for any continuous functionw, and thereforev=0.
We next show that
sup
v=1,v∈Yn
An(u,v)≥ uXn. (2.22)
DefineTn:Xn→Yn∗byTnu,v =An(u,v).Tnis a well-defined bounded linear operator andTnu=u+a(t)u. Hence
Tnu2= 1
αn
u+a(t)u2dt
= 1
αn
|u|2dt+ 1
αn
a2(t)|u|2dt+ 1
αn
a(t)(u2)dt
= 1
αn
|u|2dt+ 1
αn
a2(t)−a(t) |u|2dt+a(1)u2(1)≥ u2Xn,
(2.23)
sinceu(αn)=0,ais decreasing anda(t)≥0 for allt∈(0, 1).
All the hypotheses ofTheorem 2.3are hence satisfied and so ifF∈V∗is defined by F(v)=1
0 f v dt, then there existsu∈Xsuch that
A(u,v)=F(v), ∀v∈V. (2.24) Thususatisfies (2.16).
3. The nonlinear case
We start by recalling some well-known definitions.
Definition 3.1. LetT:X→X∗be an operator. ThenTis said to be (i) monotone ifTx−T y,x−y ≥0, for allx,y∈X;
(ii) hemicontinuous if for allx,y∈X,T(x+ty)→w Txast→0+; (iii) coercive if
xlim→∞
Tx,x
x = ∞. (3.1)
We also need the following generalization of the notion of typeM operator (for the classical definition, see [7] or [8]).
Definition 3.2. LetXbe a Banach space, letV be a linear subspace ofX, and let
A:X×V−→R (3.2)
be a function. ThenAis said to be of typeMwith respect toV if for any net{vλ}λ∈Λin V,x∈Xandv∗∈V∗;
(a)vλ→w x;
(b)A(vλ,v)→ v∗,v, for allv∈V;
(c)A(vλ,vλ)→ v∗,x, wherev∗is the extension ofv∗on the closure ofV, imply thatA(x,v)= v∗,v, for allv∈V.
Our result is the following.
Theorem 3.3. LetXbe a reflexive Banach space, letΛbe a directed set, let{Xλ}λ∈Λbe an upwards directed family of closed subspaces ofX, and letV=
λ∈ΛXλ. Suppose that
A:X×V−→R (3.3)
is a function for which the following hold:
(a)Ais of type M with respect toV; (b) limx→∞A(x,x)/x = ∞;
(c)Aλ(x,·)∈Xλ∗, for allλ∈Λand allx∈Xλ, whereAλ is the restriction of Aon Xλ×Xλ;
(d) the operatorTλ:Xλ→Xλ∗, defined byTλx,y =Aλ(x,y) for allx,y∈Xλ, is mono- tone and hemicontinuous for allλ∈Λ.
Then for eachv∗∈V∗, there existsx∈Xsuch that A(x,v)=
v∗,v, (3.4)
for allv∈V.
Proof. As in the proof ofTheorem 2.3, for eachλ∈Λ, letvλ∗=v∗|Xλ. By the Browder- Minty theorem (see [8, Theorem 26.A]), a monotone, coercive, and hemicontinuous op- erator, from a real reflexive Banach space into its dual, is onto. Thus, by (b) and (d), for
eachλ∈Λ, the operatorTλis onto and so there existsxλ∈Xλsuch that Aλxλ,y =
v∗λ,y, (3.5)
for all y∈Xλ. In particularAλ(xλ,xλ)= v∗λ,xλ, and hence by (b), we get that the net {xλ}λ∈Λ is bounded. Continuing as in the proof ofTheorem 2.3and applying the fact thatAis of typeMwith respect toV, we get the required result.
Remark 3.4. It should be noted that since a crucial point in the above proof is the existence and boundedness of the net{xλ}λ∈Λ, variants of the previous theorem could be obtained using in (b) and (d) alternative conditions corresponding to other surjectivity results.
We now applyTheorem 3.3to a singular Dirichlet problem.
Example 3.5. LetΩbe a bounded domain inRN. We consider the Dirichlet problem
−N
i=1
∂
∂xi
a(x)∂u
∂xi
+f(x,u)=0 a.e. onΩ, u=0 on∂Ω,
(3.6)
wherea∈L∞loc(Ω) and there existsc1>0 such thata(x)≥c1a.e. onΩ, and f :Ω×R→ Ris a monotone increasing (with respect to its second variable for each fixed x∈Ω) Carath´eodory function, for which there existh∈L2(Ω) andc2>0 such that
f(x,u)≤h(x) +c2|u|, ∀x∈Ω,u∈R. (3.7) We will show that if the above hypotheses onaandf hold, then problem (3.6) has a weak solution, that is, that there exists a functionu∈H01(Ω) with
Ωa(x)∇u∇v dx+
Ωf(x,u)vdx=0, ∀v∈C∞0(Ω). (3.8) To this end, letX=H01(Ω), let{Ωn}n∈Nbe an increasing sequence of open subsets of Ωsuch thatΩn⊆Ωn+1and
∞ n=1
Ωn=Ω (3.9)
and Xn=H01(Ωn), for eachn∈N. Observe that we can consider each Xn as a closed subspace ofXby extending its elements by zero outsideΩnand let
V= ∞ n=1
Xn. (3.10)
Finally, let
A:X×V−→R (3.11)
be the function defined by A(u,v)=
Ωa(x)∇u∇v dx+
Ωf(x,u)v dx. (3.12)
Bya(x)≥c1a.e. onΩ, the monotonicity off, and the growth condition (3.7), we have A(u,u)=
Ωa(x)|∇u|2dx+
Ωf(x,u)udx
=
Ωa(x)|∇u|2dx+
Ω
f(x,u)−f(x, 0) udx+
Ωf(x, 0)udx
≥c1∇u2L2(Ω)− hL2(Ω)uH01(Ω).
(3.13)
Since by the Poincar´e inequality∇uL2(Ω)is equivalent to the norm ofX, it follows that Ais coercive.
LetAn=A|Xn×Xn. Then, sincea∈L∞loc(Ω), it follows thata∈L∞(Ωn), for alln∈N. Combining this with (3.7), we have that
An(u,v)≤c(u,n)vXn, (3.14)
wherec(u,n) is a positive constant depending onnandu. So the operator
Tn:Xn−→Xn∗, (3.15)
withTnu,vXn=An(u,v), is well defined for alln∈N. Let
T1,n,T2,n:Xn−→Xn∗ (3.16)
be the operators defined by T1,nu,vXn=
Ωn
a(x)∇u∇v dx, T2,nu,vXn=
Ωn
f(x,u)v dx. (3.17) ThenT1,nis a monotone bounded linear operator. Using the monotonicity of f, it is easy to see thatT2,nis monotone. Finally, recalling that the Nemytskii operator corresponding to f is continuous (see, e.g., [8, Proposition 26.7]) and that the embedding ofXninto L2(Ωn) is compact, we have thatT2,nis hemicontinuous. ThusTn=T1,n+T2,nis mono- tone and hemicontinuous for alln∈N.
To finish the proof, letun→w uinX. Then since for allv∈V, u−→
Ωa(x)∇u∇v dx (3.18)
is a bounded linear functional and, by the continuity of the Nemytskii operator and the compactness of the embedding ofXintoL2(Ω),
Ωfx,un v dx−→
Ωf(x,u)v dx, (3.19)
for allv∈V, we get that
Aun,v −→A(u,v), ∀v∈V. (3.20) ThusAis of typeMwith respect toV. Applying nowTheorem 3.3we get that there exists u∈Xsuch thatA(u,v)=0 for allv∈V. Observing thatC0∞(Ω) is contained inV, we get thatuis the required weak solution of (3.6).
Acknowledgments
The authors would like to thank Professor A. Katavolos for pointing out an error in an earlier version of this paper and the two referees for comments and suggestions which improved both the content and the presentation of this paper.
References
[1] L. H. An, P. X. Du, D. M. Duc, and P. V. Tuoc, “Lagrange multipliers for functions derivable along directions in a linear subspace,” Proceedings of the American Mathematical Society, vol. 133, no. 2, pp. 595–604, 2005.
[2] T. L. Hayden, “The extension of bilinear functionals,” Pacific Journal of Mathematics, vol. 22, pp.
99–108, 1967.
[3] T. L. Hayden, “Representation theorems in reflexive Banach spaces,” Mathematische Zeitschrift, vol. 104, no. 5, pp. 405–406, 1968.
[4] R. E. Megginson, An Introduction to Banach Space Theory, vol. 183 of Graduate Texts in Mathe- matics, Springer, New York, NY, USA, 1998.
[5] S. Banach, Th´eorie des Op´erations Lin´eaires, Monografje Matematyczne, Warsaw, Poland, 1932.
[6] J. Dieudonn´e, “La dualit´e dans les espaces vectoriels topologiques,” Annales Scientifiques de l’ ´Ecole Normale Sup´erieure. Troisi`eme S´erie, vol. 59, pp. 107–139, 1942.
[7] H. Brezis, “ ´Equations et in´equations non lin´eaires dans les espaces vectoriels en dualit´e,” Annales de l’Institut Fourier. Universit´e de Grenoble, vol. 18, no. 1, pp. 115–175, 1968.
[8] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B, Springer, New York, NY, USA, 1990.
Dimosthenis Drivaliaris: Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Street, 82100 Chios, Greece
Email address:[email protected]
Nikos Yannakakis: Department of Mathematics, School of Applied Mathematics and Natural Sciences, National Technical University of Athens, Iroon Polytexneiou 9, 15780 Zografou, Greece Email address:[email protected]
