1 Introduction

The goal of this note is to prove existence and uniqueness of solutions for some classes of elliptic problems set in unbounded domains.More precisely ifΩis an unbounded domain in R n,let

be an n × n matrix with entries a ij ∈ L∞(R n)for i,j=1,···,n,which satisfies the usual ellipticity condition

and

Here “·” denotes the usual Euclidean product in R n,|·|the Euclidean norm,λ and Λ some positive constants.

If f denotes some distribution onΩwe would like to consider for instance problems of the type

where is the boundary ofΩ which is split into two parts where we impose Dirichlet or Neumann boundary conditions,νdenotes the outward unit normal to∂Ωsupposed to be possibly smooth.We refer to[11,13–14]for the classical notation and results on Sobolev spaces.

It is clear thatthen,using the arguments used for for(see(2.7)),leads to

Note that we made the choice of uniformizing the constantΛappearing here and in(1.2).This can be done w.l.o.g.at the expense of choosing this constant bigger.

In a bounded domain f ∈ H−1(Ω)allows to solve(1.3)relatively easily.Unfortunately whenΩ is unbounded many simple functions fail to belong to H−1(Ω)as it is the case for the constant functions(see for instance[4,9]).Thus,in this case,some new techniques have to be developed.

These kinds of problems were attacked by the Russian school in the past decades.For instance one will find in[17],(see also[12,16]),some technique of resolution of(1.3)in the distributional sense in the case,where

with p>1,a(x)≥λ>0 under the boundary conditions below.Our approach is different and allows for instance the case

Suppose that there exists some constant that w.l.o.g.we can denote byλsuch that forℓlarge enough

Let us introduce further notation.Ifωis a bounded,convex open set of R n containing 0 forℓ>0 we denote byΩℓthe set

Let Vℓ denote the closed subspace of H 1(Ωℓ)defined as

Moreover one has for some positive constants C andβindependent ofℓ

Forthe dual of Vℓand forℓlarge enough there exists a unique uℓsolution to

The hydraulic natural frequency of the EHA system is

where denotes the duality bracket betweenand Vℓ.

To see this,one sets

First note that for u∈Vℓone has

and thus the right-hand side integral in(1.11)is well defined.It is then easy to seethat A defines a monotone operator frominto (see below).Moreover this operator is hemicontinuous and coercive.Indeed to see this last point note that for all u,u0∈Vℓone has

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沿海景观带设计一般都是以人为主体、环境为载体来实现功能，它不仅是人们休闲娱乐的场所，也是在为更多的生物提供生存、繁衍的基本空间。在琅岐东南沿海岸线规划的过程中，需依托当地的生态环境，在不打破该地区生态平衡的情况下保护生态平衡，又需立足于社会经济水平，其间不能超过其承载力，从而实现岛上社会、经济和环境的共同发展，共同规划，实现三者效益的最大化。在规划中，要明确方向，对该保护的生态环境就要制定相应的保护措施，对要限制的景观建设就要制定严格的发展条件，从而达到开发与保护相结合的体系来实现景观资源可持续发展。

for some δsmall enough.Note that for a ≥ 0, the norms

are equivalent in H 1(Ωℓ)(see[2]or Section 3 below).The existence of uℓ follows then from classical results(see[3,15]).

Set

with a≥0.In addition we establish an exponential rate of convergence of the approximate solutions to(1.3)toward u.Note that here we do not assumeΩbounded in one direction and we are not relying on the Poincaré inequality(see[4–8,10]).

We will give in Section 3 below some conditions on a for this to hold.

Then we can prove the following theorem.

抗滑桩处滑动面与水平面的夹角为20°，滑坡推力的水平分力为：Fh=1076×cos20°=1011.7 kN/m。滑坡体主要以黏土与粉质黏土为主，受荷段滑坡推力沿桩身按三角形分布[22-24]。因此，在受荷段底端的单位高度的滑坡推力为：

Theorem 1.1 Under the above assumptions suppose that

比如，我就喜以日常生活中常见的荷为题材作画，一是因为广大群众喜欢荷花之美，二是因为“荷”之谐音为“和”，我可以借之表达“家和万事兴”、“国和万事兴”的主题，以之表达祝福亲朋好友生活美满，祝福祖国繁荣昌盛的情感。我的作品《祥》就是为表达这样的情感而创作的。

for some positive constantσwhich will be chosen later on.Then there exists a unique u∞ such that forℓlarge enough

Vℓ could be define for instance as the closure for the H 1(Ωℓ)-norm of the space offunctions vanishing on We will suppose Vℓ equipped with the usual norm of H 1(Ωℓ).

Remark 1.1 Note that due to the second equation of(1.18)u∞solves the first equation of(1.11)in the distributional sense.

2 Proof of Theorem 1.1

We first will need the following lemma.

Lemma 2.1 uℓ is a Cauchy “sequence”.

Proof Wedo not preciseherein what space uℓis a Cauchy“sequence”—notethatit will be clear later on.We assume thatℓis large enough in such a way that(1.16)holds.Let r ∈ [0,1].We are going to estimate first uℓ− uℓ+r.

Let R be such that

where B(0,R)denotes the Euclidean ball in R n with center 0 and radius R.Set forℓ1≤ ℓ−1,

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where∨stands for the maximum of two numbers and dist denotes the Euclidean distance.Clearly one has

To prove the last claim of(2.2)it is enough to show that ifthenIf not,for some y∈ωone has

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This implies by(2.1)that where z∈ ω.It follows that

by the convexity ofω.This completes the proof of the last claim of(2.2).

Since clearly using the equation of(1.10)forℓand ℓ+r we get

Using(2.2)we obtain

whereThus by(1.4),

Using again(2.2)–(2.3)and(1.16),we derive,thanks to the Young Inequality,

This can be written as

and thus for anyℓ1≤ ℓ−1 we get

where

Let us denote by[]the integer part of a number.Choosing forlarge enough and iterating the inequality above-times we get

passing to the limit inℓwe obtain the two first properties of(1.18).Next,going back to(2.13)written in replacingℓby 2ℓwe get for some other constant C

Sincethis leads to

We would like now to estimate the right-hand side of(2.7).Taking v=uℓin(1.10)leads to

Thus by(1.16)–(1.17)for some constant C one has

From this it follows that

如果仅仅是着力描写越战的惨烈和残酷，那么充其量不过是在累牍的越战题材小说中徒增一本可有可无的作品罢了。是的，在倾颓的焦土之上，世界现出了它的创伤与皱纹，它一下子老了，并且关心起道德，追问着为什么不能在时间之河中逆行，把我爱过的、非常想过的一种生活还给这个世界。

where C is independent ofℓ.Going back to(2.7)we obtain

We choose thenσsmall enough such that

to get for

for some constant C independent ofℓ.Recall that the norm in Vℓ is just the induced H 1(Ωℓ)-norm.

The estimate above holds for any r∈[0,1].For any t>0 one deduces then by the triangular inequality

目前，在造纸工艺中所选择的纤维素原料类型较多，包括杂草纤维如龙须草、荻、苇，农业秸秆纤维如玉米秆、棉秆[6]等，也有采用麻类[7-9]、木材、废纸等纤维素原料[10-11]。不同类型的纤维直接影响最终产品的性能和价格，如草纤维和农业秸秆纤维，不但来源丰富而且价格便宜，但是纤维强度较低，不利于提高增强纸地膜的机械强度，不利于机械化操作。麻类纤维的纤维长度和强度均好于普通秸秆纤维，但是原材料来源和成本是限制其推广应用的基本因素。

结构性制约因素方面，缺少时间是女性休闲的极大障碍，这是由于家庭和工作的双重责任挤占了大量休闲时间。Harrington M和Dawson D［26］对比家庭主妇、全职和兼职女性的休闲制约状况，家庭主妇的制约主要来自个人，如技能缺乏、对体型不自信、休闲意识淡薄等;而全职女性主要来自结构性制约，如时间缺乏、有其他安排、工作责任等;兼职女性介于二者之间，能更好地兼顾工作、家庭和休闲。此外，在男权社会背景下，不对等的休闲机会、缺少设施或适宜的项目、缺少经济资源也是导致女性不参与的重要原因［27，31-33］。人际间制约方面，由于人们通常认为女性更善于人际交往，较少面临缺乏同伴/朋友的难题［34］。

Thus for any we see that uℓis a Cauchy sequence in H 1(Ωℓ0).This completes the proof of the lemma.

End of the Proof of Theorem 1.1 Let us fix ℓ0.Forℓlarge enough uℓis a Cauchy sequence in H 1(Ωℓ0)and thus converges toward some u∞ ∈ Vℓ0(see(1.9)).Since by(1.10),

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Letting t→∞it comes

and thus by(2.8)

for some constant C.This completes the proof of(1.8).Passing to the limit in t in(2.13)leads to(1.19).To show now that the solution to(1.18)is unique ifis another solution one has

We make now more precise our assumptions on β —a Carathéodory function— such that for some a∈ L∞(Ω),a ≥ 0,a 6≡ 0,Λ >0 one has

Using the last property of(1.18)and(2.10)one deduces easily thatThis completes the proof of the theorem.

Remark 2.1 A priori u∞ depends on the choice ofω.However whenσis chosen small enough it leads to the same solution for two differentω.Indeed suppose thatω andω′are two bounded open convex subsets of R n containing 0.For some positive constant c1 one has

Suppose now R small enough in such a way that B(0,R)is included inω andω′(see(2.1)).Then Theorem 1.1 is true for σ < σ0(ω)where σ0(ω)is some constant depending on ω (see(2.10)).Then choosing

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one gets solutions to(1.18)corresponding to ω,ω′respectively.Let us denote by Vℓ(ω)(respectivelythe spaces Vℓ(respectivelycorresponding toω.Due to the inclusion above one has

制版过程的主要污染源于化学蚀刻剂。随着半导体与电路板行业的兴起，成熟的金属蚀刻技术及污染处理方案，为我们探讨铜版画的化学制版方法提供了参考。作为为硝酸的替代材料，金属盐蚀刻剂逐步应用于铜版画的制版，当然，它们也存在重金属污染的隐患。而新兴的无酸电解制版法，则代表了无毒制版的发展趋势。

and thus by(1.18)corresponding toω andω′one has

Then sincefor any one deduces as above the inequality(2.19)and the equality of u∞ andfollows.Diff erent choices ofωmight be useful for computing approximate solutions since a cube is more simple to discretise than a ball or an ellipse.Note that if(1.17)is replaced by

for some positive constantγ then the solution u∞ obtained above is independent ofω,the third condition of(2.18)being replaced by

3 Remarks and Applications

If a(x)≥λ>0 or,more precisely,a(x)≥c>0 for a constant c that without loss of generality we can take equal toλ,then clearly it follows that(1.16)holds and thus Theorem 1.1 applies.

Note that in constructing a solution to(1.18)one can possibly consider only the ℓ’s such thatℓ∈ N.Then—this is common practice in numerical analysis—suppose thatΩℓcan be covered by similar triangles,rectangles,such that on each of them one has

for some constantsµ and ǫ.Then one can show(see[1])that there exists δ= δ(µ,ǫ)such that

One should notice that the third condition in(1.18)is necessary in order to be able to state an existence and uniqueness result in all generality.Indeed suppose for instance thatΩis a domain in R2 containing the strip,

Summing up on the different Q i allows to get(1.16).Such a situation arises for instance in the case of a periodic a.We refer to[4]for details.

It is clear then that(1.16)holds.To convince the reader by a more simple case suppose that on each Q=Q i coveringΩℓthe function a is the same up perhaps to a rigid motion.Then on each Q i one has for someδ,

Suppose in addition that

i.e.,the principal part of the operator is the usual Laplace operator.Then,since a vanishes on S,it is clear that for any n the function defined by

satisfies

where∂S denotes the boundary of S.Thus adding to a solution to(1.18)a combination of these v n’s would lead to another solution to the first part of(1.18)in such a way that uniqueness is lost—even though(1.16)could be satisfied(see above).Of course for such solutions the third property of(1.18)is not satisfied.We refer the reader to[3]for the connection betweenσand λ1.

Note that the assumption(1.6)can be relaxed at the expense of changingβ(x,u)into β(x,u)−β(x,0)and f into f−β(x,0).

In the case of f=0 Theorem 1.1 provides a Liouville type result,namely,if u is a function satisfying(1.18)with f=0 then u=0.This is clear since f=0 satisfies(1.17)and then the only solution to(1.18)is 0.

Theorem 1.1 could be extended to nonlinear operators when the operator

is replaced by an operator of the type

equipped with the ad hoc structural assumptions.

Acknowledgements The author would like to thank the referee for some constructive remarks which allow him to improve this paper.This work was performed when the author visiting the City University of Hong Kong and the USTC in Hefei.He is very greatful to these institutions for their support.This article was also written during a part time employment at the S.M.Nikolskii Mathematical Institute of RUDN University,6 Miklukho-Maklay St,Moscow,117198.

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